diff --git a/.gitattributes b/.gitattributes index 28df5f900b358436f0267334b3e3e9af33f917ba..d7653e8bc31e04df8f2656e55f5c23a1aaf394d1 100644 --- a/.gitattributes +++ b/.gitattributes @@ -53,3 +53,4 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text *.jpg filter=lfs diff=lfs merge=lfs -text *.jpeg filter=lfs diff=lfs merge=lfs -text *.webp filter=lfs diff=lfs merge=lfs -text +2012/finals/possible_medians.in filter=lfs diff=lfs merge=lfs -text diff --git a/2012/finals/linsane_phone_numbers.html b/2012/finals/linsane_phone_numbers.html new file mode 100644 index 0000000000000000000000000000000000000000..94712b183e849dc84610bd4d8b763d18a01b44bc --- /dev/null +++ b/2012/finals/linsane_phone_numbers.html @@ -0,0 +1,63 @@ +
+As the Jeremy Lin sensation goes on, Roger, who is a geek and a super fan of Jeremy Lin, decides his new cell phone number must be +"Linsane". More specifically, he wants his new phone number to satisfy:
+ +
+1) Adjacent sum:
+There is at least one occurrence in the phone number of three adjacent digits summing to x, where x is Lin's
+jersey number at New York Knicks.
+2) Diversity:
+There are at least y different values of the digits used in the phone number, where y is Lin's jersey number at Golden
+State Warriors.
+3) Neighboring difference:
+There is at least one pair of neighboring digits whose difference is equal to z, where z is Lin's jersey number at Harvard.
+A phone number with length n contains n digits. Each digit is in the range from 0 to 9, except that the first
+digit must be non-zero.
+
+A phone number is called "linsane" if it satisfies the three constraints listed above.
+For phone numbers with a given length n, Roger wonders how many "linsane" phone
+numbers exist.
+He also wants to find out the "most linsane" phone number among them.
+For a given length, the "most linsane" phone number is a "linsane" phone number that has the biggest "linsanity measurement" among them.
+"Linsanity measurement" is defined as
+ , where n is the number of digits and di is the i-th digit in the phone number.
+If there is a tie on such measurement, choose the one whose median of the digits is largest; and if there is still a tie, choose the largest phone number.
+Median is the (n+1)/2-th smallest digit if n is odd, or the average of the (n/2)-th and (n/2+1)-th digit if n is even.
+For example, the linsanity measurement of number 78969251 is equal to (15*9)%8 + (17*6)%8 + (15*9)%8 + (15*2)%8 +(11*5)%8 + (7*1)%8 = 40 with its median equal to 6.5.
+
The first line contains a positive integer T, the number of test cases. T test cases follow.
+Each test case is a single line and contains exactly four integers separated by single white space: n x y z, where n is the length of the phone number, x is Lin's jersey number at New York Knicks, y is Lin's jersey number at Golden State Warriors and z is Lin's jersey number at Harvard.
+(x,y and z are not necessarily 17, 7 and 4 in another parallel universe.)
+
+
+3 ≤ n ≤ 20
+0 ≤ x ≤ 27
+0 ≤ y ≤ 10
+0 ≤ z ≤ 9
+1 ≤ T ≤ 15
+Among the T test cases, there will be no more than 5 test cases with n >12.
+
+
+
+For each of the test cases numbered in order from 1 to T, output "Case #", followed by the case number, +followed by ": ", followed by the number of possible "linsane" phone numbers mod 1018 for the given length for that case, +and then a single space " " followed by the "most linsane" phone number for the given length or -1 if no "linsane" phone number exists +for the given length. +
+ + diff --git a/2012/finals/linsane_phone_numbers.in b/2012/finals/linsane_phone_numbers.in new file mode 100644 index 0000000000000000000000000000000000000000..853a8b305a7a7ca64f7a0c4ab4855ce873af809d --- /dev/null +++ b/2012/finals/linsane_phone_numbers.in @@ -0,0 +1,16 @@ +15 +6 17 7 4 +7 17 7 4 +8 17 7 4 +9 10 5 2 +20 15 10 1 +8 0 0 5 +12 14 4 8 +19 27 10 9 +20 9 0 9 +20 10 3 5 +12 13 1 9 +8 13 7 9 +8 17 6 3 +12 3 3 4 +20 16 6 6 diff --git a/2012/finals/linsane_phone_numbers.md b/2012/finals/linsane_phone_numbers.md new file mode 100644 index 0000000000000000000000000000000000000000..0425831d0c508ce0ffb46371f4ba2fa6571ebe60 --- /dev/null +++ b/2012/finals/linsane_phone_numbers.md @@ -0,0 +1,68 @@ +As the Jeremy Lin sensation goes on, Roger, who is a geek and a super fan of +Jeremy Lin, decides his new cell phone number must be "Linsane". More +specifically, he wants his new phone number to satisfy: + +1) Adjacent sum: +There is at least one occurrence in the phone number of three adjacent digits +summing to **x**, where **x** is Lin's jersey number at New York Knicks. + +2) Diversity: +There are at least **y** different values of the digits used in the phone +number, where **y** is Lin's jersey number at Golden State Warriors. + +3) Neighboring difference: +There is at least one pair of neighboring digits whose difference is equal to +**z**, where **z** is Lin's jersey number at Harvard. + +A phone number with length **n** contains **n** digits. Each digit is in the +range from 0 to 9, except that the first digit must be non-zero. +A phone number is called "linsane" if it satisfies the three constraints +listed above. + +For phone numbers with a given length **n**, Roger wonders how many "linsane" +phone numbers exist. +He also wants to find out the "most linsane" phone number among them. +For a given length, the "most linsane" phone number is a "linsane" phone +number that has the biggest "linsanity measurement" among them. +"Linsanity measurement" is defined as ![](http://rogeryu.com/Facebook_Hacker_C +up_2012_Final_Round_Problem_Linsanity.gif), where **n** is the number of +digits and **di** is the **i**-th digit in the phone number. +If there is a tie on such measurement, choose the one whose median of the +digits is largest; and if there is still a tie, choose the largest phone +number. +Median is the **(n+1)/2**-th smallest digit if **n** is odd, or the average of +the **(n/2)**-th and **(n/2+1)**-th digit if **n** is even. For example, the +linsanity measurement of number 78969251 is equal to (15*9)%8 + (17*6)%8 + +(15*9)%8 + (15*2)%8 +(11*5)%8 + (7*1)%8 = 40 with its median equal to 6.5. + +### Input + +The first line contains a positive integer **T**, the number of test cases. +**T** test cases follow. + +Each test case is a single line and contains exactly four integers separated +by single white space: **n x y z**, where **n** is the length of the phone +number, **x** is Lin's jersey number at New York Knicks, **y** is Lin's jersey +number at Golden State Warriors and **z** is Lin's jersey number at Harvard. +(**x**,**y** and **z** are not necessarily 17, 7 and 4 in another parallel +universe.) + +### Constraints + +3 ≤ **n** ≤ 20 +0 ≤ **x** ≤ 27 +0 ≤ **y** ≤ 10 +0 ≤ **z** ≤ 9 +1 ≤ **T** ≤ 15 +Among the **T** test cases, there will be no more than 5 test cases with **n** +>12. + +### Output + +For each of the test cases numbered in order from **1** to **T**, output "Case +#", followed by the case number, followed by ": ", followed by the number of +possible "linsane" phone numbers _mod_ 1018 for the given length for that +case, and then a single space " " followed by the "most linsane" phone number +for the given length or -1 if no "linsane" phone number exists for the given +length. + diff --git a/2012/finals/linsane_phone_numbers.out b/2012/finals/linsane_phone_numbers.out new file mode 100644 index 0000000000000000000000000000000000000000..add3c635ab829f1c5ca31bfffb494bd6d3077301 --- /dev/null +++ b/2012/finals/linsane_phone_numbers.out @@ -0,0 +1,15 @@ +Case #1: 0 -1 +Case #2: 97682 9732485 +Case #3: 3657799 78969251 +Case #4: 231355703 435255828 +Case #5: 230877663780393592 90299299274965943891 +Case #6: 173063 50009699 +Case #7: 167994928670 291147255255 +Case #8: 4953342932213760 9029519993336272438 +Case #9: 613283549076340541 20929929929927243897 +Case #10: 173674046759688824 17229929929929929929 +Case #11: 87748951947 901652552557 +Case #12: 1111374 90727456 +Case #13: 11883082 78969259 +Case #14: 51212574331 102552552551 +Case #15: 559474244734633752 20929929929929927493 diff --git a/2012/finals/maximal_multiplicative_order.html b/2012/finals/maximal_multiplicative_order.html new file mode 100644 index 0000000000000000000000000000000000000000..a68bc1ade393e4add35e013539145f6c87aaa16f --- /dev/null +++ b/2012/finals/maximal_multiplicative_order.html @@ -0,0 +1,25 @@ +Let's start with some notations and definitions. Let m be the fixed positive integer.
+ +Now your task is to find for the given positive integers L and R such that L ≤ R the product +
The first line contains a positive integer T, the number of test cases. T test cases follow. The only line of each test case contains two space separated positive integers L and R.
+ +For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by the value of required product modulo 109 + 7.
+ +
+1 ≤ T ≤ 20
+1 ≤ L ≤ R ≤ 1012
+R − L ≤ 500000
+For this problem, a possible median is defined as: The median value of a set containing an odd number of values. -or- One of all the integers between, but not including, the two median values of a set containing an even number of values.
Examples: 3 is the only possible median of the set (1, 5, 3, 2, 4). 2, 3 and 4 are all of the possible median values for the set (5, 1). There
+are no possible median values for the set (1, 2, 3, 4), since there are no integers between 2 and 3.
+
+
+ +You are given an array A of N unique nonnegative integers, all of which are < N (i.e. a permutation of the numbers 0...N-1), where 3 ≤ N ≤ 200,000. You are + also given a length L (1 ≤ L ≤ N). Additionally, you are given Q (1 ≤ Q ≤ 200,000) queries, each containing a number x (0 ≤ x < N) +and two indices i and j (0 ≤ i < j ≤ N). + +
+ +Each query returns TRUE if and only if there exists at least one subrange, of at least L elements, of the range A[i]...A[j-1] (where the array indices start at 0), with x as a possible median. In other words, the answer is TRUE precisely when there exist a, b with i ≤ a < b < j with b - a ≥ L - +1 and with A[a]...A[b] having x as a possible median. + +
+ +You wish to determine the number of queries which will return TRUE. + + +
+ +The first line contains a positive integer T (1 ≤ T ≤ 20), the number of test cases. T test cases follow. + +
+The first line of each test case consists of a single integer, N. The next line consists of N space-separated integers, the A[i]. The line after that contains two space-separate +d integers, L and Q. + +
+ +Each of the remaining Q lines in the test case contains three space-separated integers, x, i and j. + +
+ +These inputs are all integers, and will be input in decimal form, with no leading zeros, and no decimal points. + +
+ +For each of the test cases numbered in order from 1 to T, output "Case #", followed by the case number, followed by ": ", followed by a single integer, the number of passing queries. + +
+ +These outputs are all integers, and must be output in decimal form, with no leading zeros, and no decimal points. + diff --git a/2012/finals/possible_medians.in b/2012/finals/possible_medians.in new file mode 100644 index 0000000000000000000000000000000000000000..36406286aab343f8cc98125be63483a7b60fa599 --- /dev/null +++ b/2012/finals/possible_medians.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c085affc10aec293048ca59e388a6f64c4fe58880a97dc78f3bc8863bdd17ec +size 33334290 diff --git a/2012/finals/possible_medians.md b/2012/finals/possible_medians.md new file mode 100644 index 0000000000000000000000000000000000000000..61e0e7866e395cb1e5c7039dd4cae1fa30588c3a --- /dev/null +++ b/2012/finals/possible_medians.md @@ -0,0 +1,46 @@ +For this problem, a _possible median_ is defined as: The median value of a set +containing an odd number of values. -or- One of all the integers between, but +not including, the two median values of a set containing an even number of +values. +**Examples:** 3 is the only possible median of the set (1, 5, 3, 2, 4). 2, 3 and 4 are all of the possible median values for the set (5, 1). There are **no** possible median values for the set (1, 2, 3, 4), since there are no integers between 2 and 3. + +You are given an array **A** of **N** unique nonnegative integers, all of +which are < **N** (i.e. a permutation of the numbers 0...**N-1**), where 3 ≤ +**N** ≤ 200,000. You are also given a length **L** (1 ≤ **L** ≤ **N**). +Additionally, you are given **Q** (1 ≤ **Q** ≤ 200,000) queries, each +containing a number **x** (0 ≤ **x** < N) and two indices **i** and **j** (0 ≤ +**i** < **j** ≤ **N**). + +Each query returns TRUE if and only if there exists at least one subrange, of +at least **L** elements, of the range **A[i]**...**A[j-1]** (where the array +indices start at 0), with **x** as a possible median. In other words, the +answer is TRUE precisely when there exist **a**, **b** with **i** ≤ **a** < +**b** < **j** with **b** \- **a** ≥ **L** \- 1 and with **A[a]**...**A[b]** +having **x** as a possible median. + +You wish to determine the number of queries which will return TRUE. + +### Input + +The first line contains a positive integer **T** (1 ≤ **T** ≤ 20), the number +of test cases. **T** test cases follow. + +The first line of each test case consists of a single integer, **N**. The next +line consists of **N** space-separated integers, the **A[i]**. The line after +that contains two space-separate d integers, **L** and **Q**. + +Each of the remaining **Q** lines in the test case contains three space- +separated integers, **x**, **i** and **j**. + +These inputs are all integers, and will be input in decimal form, with no +leading zeros, and no decimal points. + +### Output + +For each of the test cases numbered in order from 1 to T, output "Case #", +followed by the case number, followed by ": ", followed by a single integer, +the number of passing queries. + +These outputs are all integers, and must be output in decimal form, with no +leading zeros, and no decimal points. + diff --git a/2012/finals/possible_medians.out b/2012/finals/possible_medians.out new file mode 100644 index 0000000000000000000000000000000000000000..b26a629e34a6a968c35e8cc6502eb913e4b023f6 --- /dev/null +++ b/2012/finals/possible_medians.out @@ -0,0 +1,17 @@ +Case #1: 2 +Case #2: 1 +Case #3: 3 +Case #4: 1 +Case #5: 1 +Case #6: 2526 +Case #7: 1256 +Case #8: 1505 +Case #9: 3456 +Case #10: 6248 +Case #11: 32113 +Case #12: 28192 +Case #13: 44163 +Case #14: 107471 +Case #15: 73719 +Case #16: 94393 +Case #17: 48396 diff --git a/2012/quals/alphabet_soup.html b/2012/quals/alphabet_soup.html new file mode 100644 index 0000000000000000000000000000000000000000..042aaa150e55bfebfd159513dad5e2296ec47b9b --- /dev/null +++ b/2012/quals/alphabet_soup.html @@ -0,0 +1,36 @@ +
Alfredo Spaghetti really likes soup, especially when it contains alphabet +pasta. Every day he constructs a sentence from letters, places the letters into +a bowl of broth and enjoys delicious alphabet soup. +
+ +Today, after constructing the sentence, Alfredo remembered that the Facebook +Hacker Cup starts today! Thus, he decided to construct the phrase "HACKERCUP". +As he already added the letters to the broth, he is stuck with the letters he +originally selected. Help Alfredo determine how many times he can +place the word "HACKERCUP" side-by-side using the letters in his soup. +
+ +The first line of the input file contains a single integer T: the number of +test cases. T lines follow, each representing a single test case with a sequence +of upper-case letters and spaces: the original sentence Alfredo constructed. +
+ +Output T lines, one for each test case. For each case, output "Case #t: n", +where t is the test case number (starting from 1) and n is the number of times +the word "HACKERCUP" can be placed side-by-side using the letters from the +sentence. +
+ +You have encountered a new fancy online auction that offers lots of products. +You are only interested in their price and weight. We shall say that +product A is strictly preferred over product B if A costs less than B and is not +heavier (they may be of equal weight) or if A weighs less and is not more +expensive (they can have equal price). +
+ +We shall call a product A a bargain if there is no product B such that B is +better than A. Similarly, we shall call a product C a terrible deal if there +exists no product D such that C is better than D. Note that according to our +definitions, the same product may be both a bargain and a terrible deal! Only +wacky auctioneers sell such products though. +
+ +One day you wonder how many terrible deals and bargains are offered. The +number of products, N, is too large for your human-sized brain though. +Fortunately, you discovered that the auction manager is terribly lazy and +decided to sell the products based on a very simple pseudo-random number +generator. +
+ +If product i has price Pi and weight Wi, then the +following holds for product i+1: +
You carefully calculated the parameters for the generator (P1, +W1, M, K, A, B, C and D). Now you want to calculate the +number of terrible deals and bargains on the site. +
+ +The first line of the input file contains a single integer T: the number of +test cases. T lines follow, each representing a single test case with +9 space-separated integers: N, P1, W1, M, K, A, B, C and +D. +
+ +Output T lines, one for each test case. For each case, output "Case #t: a b", +where t is the test case number (starting from 1), a is the number of terrible deals +and b is the number of bargains. +
+ +We are starting preparations for Hacker Cup 2013 really early. Our first step +is to prepare billboards to advertise the contest. We have text for hundreds of +billboards, but we need your help to design them. +
+ +The billboards are of different sizes, but are all rectangular. The billboard widths and heights are all integers. We will +supply you with the size in inches and the text we want printed. We want you to +tell us how large we can print the text, such that it fits on the billboard +without splitting any words across lines. Since this is to attract hackers like +yourself, we will use a monospace font, meaning that all characters are of the +same width (e.g.. 'l' and 'm' take up the same horizontal space, as do space +characters). The characters in our font are of equal width and height, and there +will be no additional spacing between adjacent characters or adjacent rows. If you print a word on one line and print the next word on the next line, you do not need to print a space between them. +
+ +Let's say we want to print the text "Facebook Hacker Cup 2013" on a 350x100" +billboard. If we use a font size of 33" per character, then we can print +"Facebook" on the first line, "Hacker Cup" on the second and "2013" on the +third. The widest of the three lines is "Hacker Cup", which is 330" wide. There +are three lines, so the total height is 99". We cannot go any larger. +
+ +The first line of the input file contains a single integer T: the number of +test cases. T lines follow, each representing a single test case in the form "W +H S". W and H are the width and height in inches of the available space. S is +the text to be written. +
+ +Output T lines, one for each test case. For each case, output "Case #t: s", +where t is the test case number (starting from 1) and s is the maximum font +size, in inches per character, we can use. The size must be an integral number +of inches. If the text does not fit when printed at a size of 1", then output 0. +
+ ++At each point, you can move to the four immediate neighbors at a fixed speed, but since you don't want to lose the race, you are only going to take either a step to the right or to the top. +
+ ++Even though there are many ways to reach the goal while clearing the checkpoint, the race is completely pointless since it is relatively easy to figure out the shortest route. To make the race more interesting, we change the rules. + +Instead of racing to the same goal (M,N), the racers get to pick a goal (x,y) and place the checkpoint to their liking so that there are exactly S distinct shortest paths. + +
+For example, given S = 4, consider the following two goal and checkpoint configurations ++As a Hacker Cup racer, you want to figure out how to place the checkpoint and the goal so that you cannot possibly lose. Given S, find the smallest possible T, over all possible checkpoint and goal configurations, such that there are exactly S distinct shortest paths clearing the checkpoint. +
+ +Merge sort is one of the classic sorting algorithms. It divides the input array into two halves, recursively sorts each half, then merges the two sorted halves.
+ +In this problem merge sort is used to sort an array of integers in ascending order. The exact behavior is given by the following pseudo-code:
+ +function merge_sort(arr): + n = arr.length() + if n <= 1: + return arr + + // arr is indexed 0 through n-1, inclusive + mid = floor(n/2) + + first_half = merge_sort(arr[0..mid-1]) + second_half = merge_sort(arr[mid..n-1]) + return merge(first_half, second_half) + +function merge(arr1, arr2): + result = [] + while arr1.length() > 0 and arr2.length() > 0: + if arr1[0] < arr2[0]: + print '1' // for debugging + result.append(arr1[0]) + arr1.remove_first() + else: + print '2' // for debugging + result.append(arr2[0]) + arr2.remove_first() + + result.append(arr1) + result.append(arr2) + return result+ +
A very important permutation of the integers 1 through N was lost to a hard drive failure. Luckily, that sequence had been sorted by the above algorithm and the debug sequence of 1s and 2s was recorded on a different disk. You will be given the length N of the original sequence, and the debug sequence. Recover the original sequence of integers.
+ +The first line of the input file contains an integer T. This is followed by T test cases, each of which has two lines. The first line of each test case contains the length of the original sequence, N. The second line contains a string of 1s and 2s, the debug sequence produced by merge sort while sorting the original sequence. Lines are separated using Unix-style ("\n") line endings.
+ +To avoid having to upload the entire original sequence, output an integer checksum of the original sequence, calculated by the following algorithm:
+ +function checksum(arr): + result = 1 + for i=0 to arr.length()-1: + result = (31 * result + arr[i]) mod 1000003 + return result+ +
+5 ≤ T ≤ 20
+2 ≤ N ≤ 10,000
+
In the first example, N is 2 and the debug sequence is 1. The original sequence was 1 2 or 2 1. The debug sequence tells us that the first number was smaller than the second so we know the sequence was 1 2. The checksum is 994.
+ +In the second example, N is 2 and the debug sequence is 2. This time the original sequence is 2 1.
+ +In the third example, N is 4 and the debug sequence is 12212. The original sequence is 2 4 3 1.
diff --git a/2012/round1/recover_the_sequence.in b/2012/round1/recover_the_sequence.in new file mode 100644 index 0000000000000000000000000000000000000000..2ffa6a1e25fe5473056b2fee50fb014c409f3f80 --- /dev/null +++ b/2012/round1/recover_the_sequence.in @@ -0,0 +1,81 @@ +40 +2 +1 +2 +2 +4 +12212 +5 +1122211 +10 +121221212111122121212 +100 +1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 +100 +22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 +100 +121122122122212221221112122121122122222122122212112121212221221122112222112211221211221211212122112121122222222211222112211212112212112122122212122121212211212121212212112121221121212212121221121212121212212112212112211212212112121221122122122212112211211221221212112212122121122121212121121212222121212121221212211221121212121212222122112122121122122122121211221212112212121222221222211222121212121212211221121212122121121212121211221122121122112212112121212212121122121211211221121212121212122121211212211212122112212121121212212112211221122121122121212112212121211221122 +100 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diff --git a/2012/round1/recover_the_sequence.md b/2012/round1/recover_the_sequence.md new file mode 100644 index 0000000000000000000000000000000000000000..8b8e5046c3d2ab730084c87dba769a97c391b65e --- /dev/null +++ b/2012/round1/recover_the_sequence.md @@ -0,0 +1,74 @@ +Merge sort is one of the classic sorting algorithms. It divides the input +array into two halves, recursively sorts each half, then merges the two sorted +halves. + +In this problem merge sort is used to sort an array of integers in ascending +order. The exact behavior is given by the following pseudo-code: + + function merge_sort(arr): + n = arr.length() + if n <= 1: + return arr + // arr is indexed 0 through n-1, inclusive + mid = floor(n/2) + first_half = merge_sort(arr[0..mid-1]) + second_half = merge_sort(arr[mid..n-1]) + return merge(first_half, second_half) + function merge(arr1, arr2): + result = [] + while arr1.length() > 0 and arr2.length() > 0: + if arr1[0] < arr2[0]: + print '1' // for debugging + result.append(arr1[0]) + arr1.remove_first() + else: + print '2' // for debugging + result.append(arr2[0]) + arr2.remove_first() + result.append(arr1) + result.append(arr2) + return result + +A very important permutation of the integers 1 through **N** was lost to a +hard drive failure. Luckily, that sequence had been sorted by the above +algorithm and the debug sequence of 1s and 2s was recorded on a different +disk. You will be given the length **N** of the original sequence, and the +debug sequence. Recover the original sequence of integers. + +### Input + +The first line of the input file contains an integer **T**. This is followed +by **T** test cases, each of which has two lines. The first line of each test +case contains the length of the original sequence, **N**. The second line +contains a string of 1s and 2s, the debug sequence produced by merge sort +while sorting the original sequence. Lines are separated using Unix-style +("\n") line endings. + +### Output + +To avoid having to upload the entire original sequence, output an integer +checksum of the original sequence, calculated by the following algorithm: + + function checksum(arr): + result = 1 + for i=0 to arr.length()-1: + result = (31 * result + arr[i]) mod 1000003 + return result + +### Constraints + +5 ≤ **T** ≤ 20 +2 ≤ N ≤ 10,000 + +### Examples + +In the first example, N is 2 and the debug sequence is `1`. The original +sequence was 1 2 or 2 1. The debug sequence tells us that the first number was +smaller than the second so we know the sequence was 1 2. The checksum is 994. + +In the second example, N is 2 and the debug sequence is `2`. This time the +original sequence is 2 1. + +In the third example, N is 4 and the debug sequence is `12212`. The original +sequence is 2 4 3 1. + diff --git a/2012/round1/recover_the_sequence.out b/2012/round1/recover_the_sequence.out new file mode 100644 index 0000000000000000000000000000000000000000..7498e1c378f84ce4da582e94e9da2ff8cdfc5909 --- /dev/null +++ b/2012/round1/recover_the_sequence.out @@ -0,0 +1,40 @@ +Case #1: 994 +Case #2: 1024 +Case #3: 987041 +Case #4: 570316 +Case #5: 940812 +Case #6: 523359 +Case #7: 477535 +Case #8: 934263 +Case #9: 770156 +Case #10: 487200 +Case #11: 984393 +Case #12: 106718 +Case #13: 811951 +Case #14: 569818 +Case #15: 709190 +Case #16: 273515 +Case #17: 849698 +Case #18: 568298 +Case #19: 185674 +Case #20: 378672 +Case #21: 802415 +Case #22: 284922 +Case #23: 523225 +Case #24: 567769 +Case #25: 636508 +Case #26: 449921 +Case #27: 923660 +Case #28: 900872 +Case #29: 194939 +Case #30: 313933 +Case #31: 219766 +Case #32: 864609 +Case #33: 21064 +Case #34: 280883 +Case #35: 658792 +Case #36: 225962 +Case #37: 789644 +Case #38: 816281 +Case #39: 613647 +Case #40: 162643 diff --git a/2012/round1/squished_status.html b/2012/round1/squished_status.html new file mode 100644 index 0000000000000000000000000000000000000000..99cff06a954c61b890e7f8fcd0ed4cfbb8f8a20b --- /dev/null +++ b/2012/round1/squished_status.html @@ -0,0 +1,22 @@ +Some engineers got tired of dealing with all the different ways of encoding status messages, so they decided to invent their own. In their new scheme, an encoded status message consists of a sequence of integers representing the characters in the message, separated by spaces. Each integer is between 1 and M, inclusive. The integers do not have leading zeroes. Unfortunately they decided to compress the encoded status messages by removing all the spaces! +
++Your task is to figure out how many different encoded status messages a given compressed status message could have originally been. Because this number can be very large, you should return the answer modulo 4207849484 (0xfaceb00c in hex). +
++For example, if the compressed status message is "12" it might have originally been "1 2", or it might have originally been "12". The compressed status messages are between 1 and 1000 characters long, inclusive. Due to database corruption, a compressed status may contain sequences of digits that could not result from removing the spaces in an encoded status message. +
++The input begins with a single integer, N, the number of compressed status messages you must analyze. This will be followed by N compressed status messages, each consisting of an integer M, the highest character code for that database, then the compressed status message, which will be a string of digits each in the range '0' to '9', inclusive. All tokens in the input will be separated by some whitespace. +
++For each of the test cases numbered in order from 1 to N, output "Case #i: " followed by a single integer containing the number of different encoded status messages that could be represented by the corresponding compressed sequence modulo 4207849484. If none are possible, output a 0. +
+
+5 ≤ N ≤ 25
+2 ≤ M ≤ 255
+1 ≤ length of encoded status ≤ 1000
+
In a certain business sector there are currently N small companies, each having just a single employee. These employees are numbered 1 through N.
+ +The business sector is about to be transformed into a monopoly. This will happen through a series of mergers, until there is only one company. A single merger involves two companies. In a merger, the president of one company becomes the direct report of the president of the other company, preserving the rest of the hierarchies of both companies.
+ +You will be given the descriptions of all mergers. Depending on how they are performed (which of the two presidents involved becomes the president of the new company), the hierarchy can of the final company can take different shapes. We want the hierarchy of the final company to be as shallow as possible. The task is to find the smallest possible number of levels in the final hierarchy.
+ +There is also a limit D on the number of direct reports any employee can have. Because of this limit, there may be only one way to accomplish a certain merger, or it might even be impossible. However, there will always be some way to accomplish all the mergers.
+ +The first line contains the number of test cases T.
+ +Each test case starts with a blank line. The next line contains two integers, N and D.
+ +Each of the following N-1 lines describes a single merger, with two integers between 1 and N. These are the employees whose companies are merging. The two employees will never already be part of the same company.
+ +The mergers must be performed in the order in which they are given.
+ +
+5 ≤ T ≤ 20
+2 ≤ N ≤ 30,000
+1 ≤ D ≤ 5,000
+The input test cases will be such that it is possible to accomplish all mergers.
+
For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by a single integer, the smallest number of levels in the final hierarchy.
+ +In the first example, we have N=3 and D=2. The first merger happens between the companies of employees 1 and 2. In the resulting company we can have employee 1 as the president with 2 as his report, or vice versa. Next this company merges with the company of employee 3. If we have employee 3 become the president, the hierarchy will be a chain 3-1-2 or 3-2-1. If 1 or 2 become the president, that president will have the other two employees as direct reports. This last hierarchy has two levels.
+ diff --git a/2012/round2/monopoly.in b/2012/round2/monopoly.in new file mode 100644 index 0000000000000000000000000000000000000000..282cd4f142d72fe112f7fa3217d3ccebbcd15d10 --- /dev/null +++ b/2012/round2/monopoly.in @@ -0,0 +1,44008 @@ +20 + +3 2 +1 2 +1 3 + +3 1 +1 2 +1 3 + +4 2 +4 2 +2 1 +3 2 + +5 2 +5 2 +4 5 +2 1 +3 1 + +6 3 +6 1 +4 2 +1 4 +5 6 +3 6 + +50 49 +36 23 +36 40 +22 40 +36 11 +17 22 +22 5 +3 17 +5 28 +3 43 +46 40 +40 32 +38 46 +9 32 +1 17 +47 9 +45 28 +19 1 +49 3 +33 22 +6 40 +33 30 +31 23 +33 50 +17 25 +3 20 +24 49 +18 5 +47 26 +18 37 +21 23 +19 27 +41 5 +15 43 +24 4 +35 49 +20 10 +42 38 +1 16 +3 39 +2 41 +12 46 +43 8 +29 35 +14 22 +20 34 +48 32 +39 44 +19 13 +7 39 + +1000 2 +910 259 +910 648 +910 138 +544 259 +198 138 +648 995 +138 99 +198 840 +648 928 +481 138 +198 810 +481 341 +648 213 +481 281 +281 294 +995 80 +481 444 +810 982 +552 910 +213 893 +840 797 +840 723 +281 705 +995 542 +436 995 +340 910 +173 928 +501 542 +399 481 +328 982 +427 138 +352 995 +138 494 +501 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there are currently N small companies, each +having just a single employee. These employees are numbered 1 through N. + +The business sector is about to be transformed into a monopoly. This will +happen through a series of mergers, until there is only one company. A single +merger involves two companies. In a merger, the president of one company +becomes the direct report of the president of the other company, preserving +the rest of the hierarchies of both companies. + +You will be given the descriptions of all mergers. Depending on how they are +performed (which of the two presidents involved becomes the president of the +new company), the hierarchy can of the final company can take different +shapes. We want the hierarchy of the final company to be as shallow as +possible. The task is to find the smallest possible number of levels in the +final hierarchy. + +There is also a limit D on the number of direct reports any employee can have. +Because of this limit, there may be only one way to accomplish a certain +merger, or it might even be impossible. However, there will always be some way +to accomplish all the mergers. + +### Input + +The first line contains the number of test cases T. + +Each test case starts with a blank line. The next line contains two integers, +N and D. + +Each of the following N-1 lines describes a single merger, with two integers +between 1 and N. These are the employees whose companies are merging. The two +employees will never already be part of the same company. + +The mergers must be performed in the order in which they are given. + +### Constraints + +5 ≤ T ≤ 20 +2 ≤ N ≤ 30,000 +1 ≤ D ≤ 5,000 +The input test cases will be such that it is possible to accomplish all +mergers. + +### Output + +For each of the test cases numbered in order from 1 to T, output "Case #i: " +followed by a single integer, the smallest number of levels in the final +hierarchy. + +### Examples + +In the first example, we have N=3 and D=2. The first merger happens between +the companies of employees 1 and 2. In the resulting company we can have +employee 1 as the president with 2 as his report, or vice versa. Next this +company merges with the company of employee 3. If we have employee 3 become +the president, the hierarchy will be a chain 3-1-2 or 3-2-1. If 1 or 2 become +the president, that president will have the other two employees as direct +reports. This last hierarchy has two levels. + diff --git a/2012/round2/monopoly.out b/2012/round2/monopoly.out new file mode 100644 index 0000000000000000000000000000000000000000..e9eca449b95c00920ab11390eb8bd7efc9dba611 --- /dev/null +++ b/2012/round2/monopoly.out @@ -0,0 +1,20 @@ +Case #1: 2 +Case #2: 3 +Case #3: 3 +Case #4: 3 +Case #5: 4 +Case #6: 2 +Case #7: 501 +Case #8: 8 +Case #9: 7 +Case #10: 7 +Case #11: 6 +Case #12: 19 +Case #13: 16 +Case #14: 13 +Case #15: 35 +Case #16: 45 +Case #17: 22 +Case #18: 15 +Case #19: 8 +Case #20: 1244 diff --git a/2012/round2/road_removal.html b/2012/round2/road_removal.html new file mode 100644 index 0000000000000000000000000000000000000000..f7e772cd92eb3519759d8f50d708d306948c616a --- /dev/null +++ b/2012/round2/road_removal.html @@ -0,0 +1,24 @@ +You are given a network with N cities and M bidirectional roads connecting these cities. The first K cities are important. You need to remove the minimum number of roads such that in the remaining network there are no cycles that contain important cities. A cycle is a sequence of at least three different cities such that each pair of neighboring cities are connected by a road and the first and the last city in the sequence are also connected by a road.
+ +The first line contains the number of test cases T.
+ +Each case begins with a line containing integers N, M and K, which represent the number of cities, the number of roads and the number of important cities, respectively. The cities are numbered from 0 to N-1, and the important cities are numbered from 0 to K-1. The following M lines contain two integers a[i] and b[i], 0 ≤ i < M, that represent two different cities connected by a road.
+ +It is guaranteed that 0 ≤ a[i], b[i] < N and a[i] ≠ b[i]. There will be at most one road between two cities.
+ +For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by a single integer, the minimum number of roads that need to be removed such that there are no cycles that contain an important city.
+ +
+1 ≤ T ≤ 20
+1 ≤ N ≤ 10,000
+1 ≤ M ≤ 50,000
+1 ≤ K ≤ N
In the first example, we have N=5 cities that are connected by M=7 roads and the cities 0 and 1 are important. We can remove two roads connecting (0, 1) and (1, 2) and the remaining network will not contain cycles with important cities. Note that in the remaining network there is a cycle that contains only non-important cities, and that there are also multiple ways to remove two roads and satisfy all conditions. One cannot remove only one road and destroy all cycles that contain important cities.
+ diff --git a/2012/round2/road_removal.in b/2012/round2/road_removal.in new file mode 100644 index 0000000000000000000000000000000000000000..ebd971c2bbee57579e749d080d7b024b20319399 --- /dev/null +++ b/2012/round2/road_removal.in @@ -0,0 +1,94485 @@ +20 +5 7 2 +0 1 +1 2 +1 4 +0 2 +2 4 +2 3 +3 4 +8 12 2 +0 2 +0 4 +0 5 +2 3 +2 7 +3 1 +3 4 +4 6 +5 6 +5 7 +6 1 +7 1 +99 3890 1 +54 3 +54 7 +54 95 +54 70 +54 43 +54 89 +54 34 +54 53 +54 19 +54 86 +54 72 +54 41 +54 15 +54 68 +54 82 +54 2 +54 6 +54 1 +54 9 +54 20 +54 97 +54 13 +54 59 +54 38 +54 75 +54 11 +54 14 +54 40 +54 83 +54 12 +54 44 +54 37 +54 63 +54 62 +54 16 +54 48 +54 52 +54 66 +54 80 +54 10 +54 76 +54 84 +54 74 +54 0 +54 51 +54 28 +54 17 +54 49 +54 4 +54 18 +54 31 +54 39 +54 77 +54 58 +54 81 +54 67 +54 98 +54 85 +54 90 +54 22 +54 61 +54 23 +54 88 +54 91 +54 25 +54 42 +54 71 +54 5 +54 45 +54 21 +54 32 +54 50 +54 8 +3 70 +3 89 +3 55 +3 64 +3 34 +3 53 +3 19 +3 86 +3 96 +3 79 +3 72 +3 41 +3 68 +3 82 +3 2 +3 6 +3 20 +3 97 +3 57 +3 13 +3 59 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8034 +766 1578 +766 4840 +766 3614 +2860 4141 +7507 7572 +4157 61 +902 2156 +2355 4463 +5235 346 +2213 2888 +1931 2894 +6980 4455 +1770 7101 +743 2247 +4463 6109 +2513 2208 +508 5373 +508 6814 +9947 3044 +7260 334 +2885 1136 +1030 5919 +334 2972 +9265 297 +6828 2850 +397 401 +397 767 +6876 1193 +5547 260 +4335 346 +8509 4147 +8713 2972 +989 2375 +6178 3316 +5652 4510 +818 3865 diff --git a/2012/round2/road_removal.md b/2012/round2/road_removal.md new file mode 100644 index 0000000000000000000000000000000000000000..b36bce1ae969e5f4ee06ef072505781930bddbd4 --- /dev/null +++ b/2012/round2/road_removal.md @@ -0,0 +1,45 @@ +You are given a network with **N** cities and **M** bidirectional roads +connecting these cities. The first **K** cities are important. You need to +remove the minimum number of roads such that in the remaining network there +are no cycles that contain important cities. A cycle is a sequence of at least +three different cities such that each pair of neighboring cities are connected +by a road and the first and the last city in the sequence are also connected +by a road. + +## Input + +The first line contains the number of test cases **T**. + +Each case begins with a line containing integers **N**, **M** and **K**, which +represent the number of cities, the number of roads and the number of +important cities, respectively. The cities are numbered from **0** to **N-1**, +and the important cities are numbered from **0** to **K-1**. The following +**M** lines contain two integers **a[i]** and **b[i]**, **0** ≤ **i** < **M**, +that represent two different cities connected by a road. + +It is guaranteed that **0** ≤ **a[i]**, **b[i]** < **N** and **a[i]** ≠ +**b[i]**. There will be at most one road between two cities. + +## Output + +For each of the test cases numbered in order from **1** to **T**, output "Case +#i: " followed by a single integer, the minimum number of roads that need to +be removed such that there are no cycles that contain an important city. + +## Constraints + +1 ≤ **T** ≤ 20 +1 ≤ **N** ≤ 10,000 +1 ≤ **M** ≤ 50,000 +1 ≤ **K** ≤ **N** + +### Example + +In the first example, we have **N**=5 cities that are connected by **M**=7 +roads and the cities **0** and **1** are important. We can remove two roads +connecting (**0**, **1**) and (**1**, **2**) and the remaining network will +not contain cycles with important cities. Note that in the remaining network +there is a cycle that contains only non-important cities, and that there are +also multiple ways to remove two roads and satisfy all conditions. One cannot +remove only one road and destroy all cycles that contain important cities. + diff --git a/2012/round2/road_removal.out b/2012/round2/road_removal.out new file mode 100644 index 0000000000000000000000000000000000000000..2e8678ecab2cf5a854c9123b52d2cfea548e7b63 --- /dev/null +++ b/2012/round2/road_removal.out @@ -0,0 +1,20 @@ +Case #1: 2 +Case #2: 4 +Case #3: 72 +Case #4: 33 +Case #5: 361 +Case #6: 398 +Case #7: 4851 +Case #8: 1176 +Case #9: 511 +Case #10: 1014 +Case #11: 1114 +Case #12: 286 +Case #13: 0 +Case #14: 1070 +Case #15: 731 +Case #16: 22 +Case #17: 3135 +Case #18: 13995 +Case #19: 4569 +Case #20: 190 diff --git a/2012/round2/sequence_slicing.html b/2012/round2/sequence_slicing.html new file mode 100644 index 0000000000000000000000000000000000000000..71f786a7a87c06393bd3ec5b03dd79debd0aab5d --- /dev/null +++ b/2012/round2/sequence_slicing.html @@ -0,0 +1,39 @@ +Let S be a sequence of N natural numbers. We can define an infinite sequence MS in the following way: +MS[k] = S[k mod N] + N * floor(k / N). +Where k is a zero based index.
+For example if the sequence S is {2, 1, 3} then MS would be {2, 1, 3, 5, 4, 6, 8, 7, 9, 11, 10, 12...}
+ +Now consider a subsequence of MS generated by picking two random indices a, b from the range [0..R] inclusive, and taking all the elements between them, that is: +MS[min(a, b)..max(a, b)]. +
If we use the same MS as in the example above and a = 2, b = 5 then our subsequence would be {3, 5, 4, 6}. +
+Your task is to calculate the probability that the selected subsequence has at least K distinct elements. a and b are selected independently and with a uniform distribution. The result should be printed as a fraction. See the "Output" section for clarification. + +
The first line of the input file contains an integer T. This is followed by T test cases, each of which has two lines.
+The first line of each test case contains three integers separated by spaces, N, K, and R.
+
+
The second line contains N space separated integers, S[0] through S[N-1].
+ +1 ≤ T ≤ 20
+1 ≤ N ≤ 2,000
+1 ≤ K ≤ R ≤ 1,000,000,000
+1 ≤ S[i] ≤ 100,000
+For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by the probability that the selected subsequence of MS has at least K distinct elements. The probability should be expressed as a fraction p/q, where p and q represent the numerator and denominator respectively and are relatively prime (that is they share no common positive divisors except 1). +
+ +If the probability is 0 or 1 output 0/1 or 1/1 respectively.
+ +In the first example there are 36 different subsequences to consider. 6 of them have only a single number, and the remaining 30 have at least 2 different numbers, so the answer is 5/6. +
++The second example is similar, but now the sequence looks like {2, 1, 5, 5, 4, 8}. There are 8 subsequences with less than 2 distinct numbers: the six single number subsequences plus (a=2, b=3) and (a=3, b=2) which both result in {5,5}. That gives a probability of (36 - 8) / 36 = 7/9. +
++The third example uses the same sequence as the second example, but now we want to have subsequences with at least 4 different numbers. All pairs of indices that have this property are: (0,4), (0, 5), (1, 5), (4, 0), (5, 0), and (5, 1). Six out of thirty six results in a probability of 1/6. +
diff --git a/2012/round2/sequence_slicing.in b/2012/round2/sequence_slicing.in new file mode 100644 index 0000000000000000000000000000000000000000..1b8eaac2dbfb7ca9ea5dce5c21899fb5661f9117 --- /dev/null +++ b/2012/round2/sequence_slicing.in @@ -0,0 +1,69 @@ +34 +3 2 5 +2 1 3 +3 2 5 +2 1 5 +3 4 5 +2 1 5 +3 5 5 +2 2 5 +10 1000 1000000 +1 2 3 4 5 6 7 8 9 10 +740 725 2416 +1973 3031 2178 758 3325 4707 4007 4509 1420 1755 3586 3738 3080 3510 702 2878 4477 990 1920 2443 141 946 216 2625 3737 3909 4721 4823 4038 243 1960 1178 3227 3065 3553 2532 3583 2142 1412 3180 2111 2553 359 4397 989 1016 4645 3702 4508 1602 660 3374 2872 3482 4644 4682 826 2120 2075 1533 1337 3246 1982 1843 2557 1300 1034 4745 1260 104 369 1916 2015 3061 2056 2900 31 3863 4124 2437 2193 3249 4084 2763 4985 675 1378 1575 3826 4539 3956 3554 1147 4430 4415 4732 4310 4771 3395 1290 3146 2973 3097 1666 4760 68 1056 2964 2482 4990 1263 302 3159 4814 1502 4428 532 4721 701 1987 3572 1906 702 2705 4936 1777 3958 4050 3282 4868 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35000 11000 84000 41000 73000 34000 12000 54000 16000 28000 42000 70000 97000 89000 35000 15000 86000 45000 29000 61000 87000 31000 3000 55000 70000 46000 54000 11000 21000 58000 72000 67000 7000 58000 42000 96000 84000 49000 53000 74000 68000 58000 55000 18000 32000 87000 34000 9000 61000 55000 43000 32000 74000 67000 55000 66000 83000 53000 82000 50000 89000 55000 60000 18000 21000 52000 10000 55000 53000 69000 63000 21000 72000 52000 69000 22000 1000 24000 52000 76000 80000 29000 28000 50000 90000 84000 19000 55000 50000 37000 98000 94000 27000 62000 93000 58000 41000 82000 52000 7000 29000 18000 2000 74000 73000 9000 37000 77000 32000 52000 96000 61000 71000 49000 33000 72000 43000 43000 65000 89000 54000 28000 74000 16000 99000 44000 6000 78000 67000 69000 18000 12000 29000 47000 77000 18000 63000 54000 68000 33000 58000 62000 53000 75000 33000 96000 2000 75000 3000 70000 88000 52000 6000 69000 15000 85000 84000 30000 10000 85000 73000 28000 82000 52000 95000 96000 40000 11000 80000 41000 3000 49000 50000 95000 3000 95000 52000 88000 34000 16000 5000 27000 40000 58000 48000 90000 71000 34000 56000 18000 18000 80000 20000 34000 7 26000 47000 95000 47000 27000 47000 12000 53000 84000 28000 67000 74000 28000 25000 36000 90000 73000 17000 52000 20000 51000 1000 95000 53000 50000 78000 34000 4000 67000 5000 86000 46000 75000 45000 50000 67000 64000 81000 85000 88000 61000 6000 66000 61000 85000 80000 99000 53000 51000 81000 50000 35000 37000 16000 40000 69000 82000 92000 81000 66000 5000 97000 13000 4000 43000 61000 28000 87000 29000 88000 39000 26000 51000 54000 85000 30000 16000 33000 98000 34000 52000 23000 54000 98000 24000 18000 84000 77000 59000 86000 64000 54000 92000 52000 63000 93000 67000 48000 81000 22000 81000 76000 82000 58000 32000 31000 99000 1000 68000 11000 1000 61000 84000 74000 3000 24000 68000 5000 86000 71000 86000 38000 13000 29000 13000 49000 96000 23000 29000 93000 73000 30000 76000 90000 15000 28000 68000 14000 75000 11000 62000 57000 28000 11000 61000 38000 14000 32000 19000 86000 45000 18000 41000 98000 53000 69000 17000 59000 73000 50000 39000 59000 23000 86000 79000 93000 51000 52000 77000 86000 69000 96000 92000 95000 19000 60000 23000 59000 18000 5000 58000 97000 76000 51000 91000 45000 29000 86000 3000 66000 83000 61000 74000 40000 57000 67000 66000 96000 41000 24000 20000 95000 30000 33000 62000 55000 24000 5000 6000 1000 54000 60000 23000 37000 85000 20000 14000 7000 29000 89000 34000 99000 28000 1000 50000 25000 31000 67000 44000 41000 97000 64000 2000 +1 1000000000 1000000000 +1 diff --git a/2012/round2/sequence_slicing.md b/2012/round2/sequence_slicing.md new file mode 100644 index 0000000000000000000000000000000000000000..cfd5b23342a19c4fdfa16a1465a31f1f5e27e254 --- /dev/null +++ b/2012/round2/sequence_slicing.md @@ -0,0 +1,64 @@ +Let **S** be a sequence of **N** natural numbers. We can define an infinite +sequence **MS** in the following way: **MS**[k] = **S**[k mod **N**] + **N** * +floor(k / **N**). Where k is a zero based index. + +For example if the sequence **S** is {2, 1, 3} then **MS** would be {2, 1, 3, +5, 4, 6, 8, 7, 9, 11, 10, 12...} + +Now consider a subsequence of **MS** generated by picking two random indices +**a**, **b** from the range [**0**..**R**] inclusive, and taking all the +elements between them, that is: **MS**[min(**a**, **b**)..max(**a**, **b**)]. + +If we use the same **MS** as in the example above and **a** = 2, **b** = 5 +then our subsequence would be {3, 5, 4, 6}. + +Your task is to calculate the probability that the selected subsequence has at +least **K** distinct elements. **a** and **b** are selected independently and +with a uniform distribution. The result should be printed as a fraction. See +the "Output" section for clarification. + +### Input + +The first line of the input file contains an integer **T**. This is followed +by **T** test cases, each of which has two lines. + +The first line of each test case contains three integers separated by spaces, +**N**, **K**, and **R**. + +The second line contains **N** space separated integers, **S**[0] through +**S**[**N**-1]. + +### Constraints + +1 ≤ **T** ≤ 20 +1 ≤ **N** ≤ 2,000 +1 ≤ **K** ≤ **R** ≤ 1,000,000,000 +1 ≤ **S**[i] ≤ 100,000 + +### Output + +For each of the test cases numbered in order from 1 to **T**, output "Case +#**i**: " followed by the probability that the selected subsequence of **MS** +has at least **K** distinct elements. The probability should be expressed as a +fraction **p**/**q**, where **p** and **q** represent the numerator and +denominator respectively and are relatively prime (that is they share no +common positive divisors except 1). + +If the probability is 0 or 1 output 0/1 or 1/1 respectively. + +### Examples + +In the first example there are 36 different subsequences to consider. 6 of +them have only a single number, and the remaining 30 have at least 2 different +numbers, so the answer is 5/6. + +The second example is similar, but now the sequence looks like {2, 1, 5, 5, 4, +8}. There are 8 subsequences with less than 2 distinct numbers: the six single +number subsequences plus (a=2, b=3) and (a=3, b=2) which both result in {5,5}. +That gives a probability of (36 - 8) / 36 = 7/9. + +The third example uses the same sequence as the second example, but now we +want to have subsequences with at least 4 different numbers. All pairs of +indices that have this property are: (0,4), (0, 5), (1, 5), (4, 0), (5, 0), +and (5, 1). Six out of thirty six results in a probability of 1/6. + diff --git a/2012/round2/sequence_slicing.out b/2012/round2/sequence_slicing.out new file mode 100644 index 0000000000000000000000000000000000000000..9015dd7982d8bbedf94812638985348d4cdec6fd --- /dev/null +++ b/2012/round2/sequence_slicing.out @@ -0,0 +1,34 @@ +Case #1: 5/6 +Case #2: 7/9 +Case #3: 1/6 +Case #4: 0/1 +Case #5: 998005995006/1000002000001 +Case #6: 2692922/5841889 +Case #7: 69911/315844 +Case #8: 11382034/14250625 +Case #9: 23677986/24611521 +Case #10: 233678/3876961 +Case #11: 1/1 +Case #12: 10566737/27602450 +Case #13: 894152/6171605 +Case #14: 631400/4531323 +Case #15: 2727724/54272689 +Case #16: 51911/46967432 +Case #17: 2084441/4456321 +Case #18: 13044037/18060050 +Case #19: 0/1 +Case #20: 1/1 +Case #21: 7406024/21669025 +Case #22: 335222/1575025 +Case #23: 4188535/6780816 +Case #24: 172484/1565001 +Case #25: 62267074/95942025 +Case #26: 10535999529895110/85209558285138169 +Case #27: 568783218408907/8314358527108732 +Case #28: 0/1 +Case #29: 416515005307519/7653652276052250 +Case #30: 1844737452390571/1890800074855561 +Case #31: 35334298477125432/35624192971900729 +Case #32: 98894340234057202/100064691043101225 +Case #33: 778741908690941/859699982029952 +Case #34: 6/1000000002000000001 diff --git a/2012/round3/divisor_function_optimization.html b/2012/round3/divisor_function_optimization.html new file mode 100644 index 0000000000000000000000000000000000000000..5d08a6bd262f0cd69082ca89c21c0ae9fcef89a9 --- /dev/null +++ b/2012/round3/divisor_function_optimization.html @@ -0,0 +1,19 @@ +Let d(N) be the number of positive divisors of positive integer N. +Consider the infinite sequence x(n) = d(n)a / nb, n = 1, 2, 3, … +where a and b are fixed positive integers. +It can be shown that this sequence tends to zero. Hence it attains its maximum. Denote it by p/q where p and q are co-prime positive integers. Your task is for given a and b find p and q modulo M = 109+7. But to keep input and output small you will be given tuples (b1; b2; a1; a2; c) and need to calculate the sum of (p mod M) for all pairs (a; b) such that b1 ≤ b ≤ b2, a1 ≤ a ≤ a2 and a ≤ c*b, and the same sum for q-values.
+ +The first line contains a positive integer T, the number of test cases. T test cases follow. The only line of each test case contains five space separated positive integers b1, b2, a1, a2 and c.
+ +For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by a space separated pair of integers: the sum of (p mod M) for all pairs (a; b) mentioned above and the sum of (q mod M) for all such pairs. Note that you need to find the sum of residues not the residue of sum (see testcase 3 as a reference).
+ +
+1 ≤ T ≤ 20
+1 ≤ b1 ≤ b2 ≤ 10,000
+1 ≤ a1 ≤ a2 ≤ 250,000
+1 ≤ c ≤ 25
+in each testcase the total number of pairs (a; b) for which the answer should be calculated does not exceed 100,000
Consider two horizontal lines and a set of N trapezoids. A trapezoid T[i] between these lines has two vertices situated on the upper line and the other two vertices on the lower line. We will denote by a[i], b[i], c[i] and d[i] the upper left, upper right, lower left and respectively lower right vertices of the trapezoid T[i]. Assume that no two trapezoids share a common vertex (meaning that all a [i] and b [i] coordinates on the upper line are different, and the same holds for the bottom line and coordinates c [i] and d [i]). A trapezoid set is called disconnected if one can separate the trapezoids in two or more groups such that no two trapezoids from different groups intersect. Determine the smallest number of trapezoids to remove, such that the remaining trapezoids form a disconnected set. If the solution does not exist, output -1.
+ +The first line of the input file contains an integer T, and this is followed by T test cases. Each test case is given in the compressed format. + +
The first line of each test case contains the number of trapezoids N, an integer K, and integer parameters X, A, B, M, p, q. In the next K lines are given integer numbers aa[i], bb[i], cc[i], dd[i]. The following code is used for generating the next random number using linear congruential generator with the starting value X and parameters A, B and modulo M: + +
+long long prior = X; +long long next() { + prior = (A * prior + B) % M; + return prior; +} ++ +The following code is used for extending the auxiliary sequences aa, bb, cc and dd: +
+for (int i = K; i < N; i++) { + aa [i] = aa [i - K] + next() % (2 * p) - p; + bb [i] = aa [i] + 1 + next() % (2 * (bb [i % K] - aa [i % K])); + cc [i] = cc [i - K] + next() % (2 * q) - q; + dd [i] = cc [i] + 1 + next() % (2 * (dd [i % K] - cc [i % K])); +} ++ +The final coordinates of the trapezoids are given by: +
+for (int i = 0; i < N; i++) { + a [i] = aa [i] * 1000000 + i; + b [i] = bb [i] * 1000000 + i; + c [i] = cc [i] * 1000000 + i; + d [i] = dd [i] * 1000000 + i; +} ++ +Note that above code generates trapezoids that satisfy all conditions of the problem, and '%' denotes the remainder of division. + +
For each of the test cases numbered in order from 1 to T, output "Case #i: " followed by a single integer, the minimum number of trapezoids that need to be removed such that the remaining trapezoids form a disconnected set.
+ +
+1 ≤ T ≤ 20
+1 ≤ N ≤ 300,000
+1 ≤ K ≤ N
+0 ≤ X, A, B ≤ 2,000,000,000
+-2,000,000,000 ≤ aa [i], bb [i], cc [i], dd [i] ≤ 2,000,000,000
+1 ≤ p, q, M ≤ 2,000,000,000
+
+
In the first example, one can remove trapezoids 5 and 6 leaving two disconnected sets with trapezoids (1, 2, 3, 4) and (7, 8).
+ +In the second example, the coordinates of ten trapezoids are +
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3 +1 2 2 4 +1 2 2 3 + +300000 1 4 3 5 223 1000 100 +-10235 22222 -3425 80003 + +250000 5 100 23 1 223 5 6 +0 6 0 3 +1000000 1000017 100000 100003 +2000000 2000031 200000 200006 +3000000 3000036 300000 300010 +4000000 4000003 400000 400003 diff --git a/2012/round3/trapezoids.md b/2012/round3/trapezoids.md new file mode 100644 index 0000000000000000000000000000000000000000..5449029368ac0beb4605a873d2c848327f16b6ae --- /dev/null +++ b/2012/round3/trapezoids.md @@ -0,0 +1,87 @@ +Consider two horizontal lines and a set of **N** trapezoids. A trapezoid +**T[i]** between these lines has two vertices situated on the upper line and +the other two vertices on the lower line. We will denote by **a[i], b[i], +c[i]** and **d[i]** the upper left, upper right, lower left and respectively +lower right vertices of the trapezoid **T[i]**. Assume that no two trapezoids +share a common vertex (meaning that all **a [i]** and **b [i]** coordinates on +the upper line are different, and the same holds for the bottom line and +coordinates **c [i]** and **d [i]**). A trapezoid set is called +**disconnected** if one can separate the trapezoids in two or more groups such +that no two trapezoids from different groups intersect. Determine the smallest +number of trapezoids to remove, such that the remaining trapezoids form a +disconnected set. If the solution does not exist, output **-1**. + +### Input + +The first line of the input file contains an integer **T**, and this is +followed by **T** test cases. Each test case is given in the compressed +format. + +The first line of each test case contains the number of trapezoids **N**, an +integer **K**, and integer parameters **X, A, B, M, p, q**. In the next **K** +lines are given integer numbers **aa[i], bb[i], cc[i], dd[i]**. The following +code is used for generating the next random number using linear congruential +generator with the starting value **X** and parameters **A, B** and modulo +**M**: + + long long prior = X; + long long next() { + prior = (A * prior + B) % M; + return prior; + } + +The following code is used for extending the auxiliary sequences **aa, bb, +cc** and **dd**: + + for (int i = K; i < N; i++) { + aa [i] = aa [i - K] + next() % (2 * p) - p; + bb [i] = aa [i] + 1 + next() % (2 * (bb [i % K] - aa [i % K])); + cc [i] = cc [i - K] + next() % (2 * q) - q; + dd [i] = cc [i] + 1 + next() % (2 * (dd [i % K] - cc [i % K])); + } + +The final coordinates of the trapezoids are given by: + + for (int i = 0; i < N; i++) { + a [i] = aa [i] * 1000000 + i; + b [i] = bb [i] * 1000000 + i; + c [i] = cc [i] * 1000000 + i; + d [i] = dd [i] * 1000000 + i; + } + +Note that above code generates trapezoids that satisfy all conditions of the +problem, and '%' denotes the remainder of division. + +## Output + +For each of the test cases numbered in order from **1** to **T**, output "Case +#i: " followed by a single integer, the minimum number of trapezoids that need +to be removed such that the remaining trapezoids form a disconnected set. + +## Constraints + +1 ≤ **T** ≤ 20 +1 ≤ **N** ≤ 300,000 +1 ≤ **K** ≤ **N** +0 ≤ **X, A, B** ≤ 2,000,000,000 +-2,000,000,000 ≤ **aa [i], bb [i], cc [i], dd [i]** ≤ 2,000,000,000 +1 ≤ **p, q, M** ≤ 2,000,000,000 + +### Examples + +In the first example, one can remove trapezoids 5 and 6 leaving two +disconnected sets with trapezoids (1, 2, 3, 4) and (7, 8). + +In the second example, the coordinates of ten trapezoids are + + 1000000 4000000 3000000 5000000 + 7000001 8000001 2000001 6000001 + 2000002 7000002 2000002 5000002 + 5000003 7000003 2000003 6000003 + 4 6000004 2000004 3000004 + 4000005 5000005 3000005 8000005 + -999994 6 6 2000006 + 4000007 6000007 1000007 7000007 + -999992 8 -999992 2000008 + 5000009 6000009 9 7000009 + diff --git a/2012/round3/trapezoids.out b/2012/round3/trapezoids.out new file mode 100644 index 0000000000000000000000000000000000000000..d27389db013235d57a15c5b6b536da651bc7d28e --- /dev/null +++ b/2012/round3/trapezoids.out @@ -0,0 +1,30 @@ +Case #1: 2 +Case #2: 6 +Case #3: 50002 +Case #4: 61 +Case #5: -1 +Case #6: -1 +Case #7: 1 +Case #8: 25 +Case #9: 161 +Case #10: 988 +Case #11: 1 +Case #12: 100002 +Case #13: 32183 +Case #14: 49898 +Case #15: 120004 +Case #16: 498 +Case #17: 1 +Case #18: 82 +Case #19: 16 +Case #20: 14213 +Case #21: 838 +Case #22: 3291 +Case #23: 47541 +Case #24: 1 +Case #25: 35958 +Case #26: 3 +Case #27: 2582 +Case #28: 382 +Case #29: 1 +Case #30: 0 diff --git a/2012/round3/unfriending.html b/2012/round3/unfriending.html new file mode 100644 index 0000000000000000000000000000000000000000..46ca612662daa241f811b22226f59674b5d5e580 --- /dev/null +++ b/2012/round3/unfriending.html @@ -0,0 +1,44 @@ +
+It's time to clean up your friends list after friending too many people on Facebook over the winter. +The set of your friends' user ids is F = {x0, x1, ..., xn-1}. Each friend from F is on zero or more friend lists that you use to organize your friends. There are M friend lists, c0, c1, ..., cm-1. + +You can unfriend (remove from F) some of your friends, but because you want to stay in touch with people, you can only unfriend one person from each friend list. You may also choose not to unfriend anyone from friend list. Any friend that isn't on a friend list cannot be unfriended.
+ ++Your goal is to find the maximum possible distance between the two friends whose user ids are the closest after you are done unfriending people. The distance between user id x and user id y is defined as abs(x-y). +
+ +The first line contains a positive integer T, the number of test cases. T test cases follow.
+ +The first line of each test case contains the number of friends N, and number of friend lists M, separated by a space. The second line contains x0 and integer parameters a, b, and p, separated by spaces. You must use these to generate the remaining numbers x1, …, xn-1 according to this formula: + +
+xi = (xi-1 * a + b) mod p ++ +The next M lines of each test case define the friend lists for that test case. Each line consists of the following integers, separated by spaces: size, y0, a, b. The size is the number of friends in the friend list. y0 is the index in F of the first friend in the list. You must generate the remaining friends in the friend list y1, …, yn-1 according to the following formula: + +
+yi = (yi-1 * a + b) mod n ++ +If a friend list contains more than one of a given index, consider it to only contain that index once. + + +
+1 ≤ T ≤ 20
+2 ≤ n ≤ 50 000
+0 ≤ m ≤ 1 500
+0 ≤ sum of sizes of all friend lists ≤ 1 000 000
+Numbers used in generators:
+0 < a, p < 230
+0 ≤ b < 230
+
+
+For each of the test cases numbered in order from 1 to T, output "Case #", followed by the case number, followed by ": ", followed by the maximum possible distance between the two closest user ids that you are still friends with for that case. +
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2764366 +666 48513 127807389 905128041 +666 13072 461245297 938139162 +666 5060 470027288 138606636 diff --git a/2012/round3/unfriending.md b/2012/round3/unfriending.md new file mode 100644 index 0000000000000000000000000000000000000000..f5a103cf5e6c8842b2c41496b3eb24d57a1b905a --- /dev/null +++ b/2012/round3/unfriending.md @@ -0,0 +1,57 @@ +It's time to clean up your friends list after friending too many people on +Facebook over the winter. The set of your friends' user ids is **F** = +{**x0**, **x1**, ..., **xn-1**}. Each friend from **F** is on zero or more +friend lists that you use to organize your friends. There are **M** friend +lists, **c0**, **c1**, ..., **cm-1**. You can unfriend (remove from **F**) +some of your friends, but because you want to stay in touch with people, you +can only unfriend one person from each friend list. You may also choose not to +unfriend anyone from friend list. Any friend that isn't on a friend list +cannot be unfriended. + +Your goal is to find the maximum possible distance between the two friends +whose user ids are the closest after you are done unfriending people. The +distance between user id **x** and user id **y** is defined as +abs(**x**-**y**). + +### Input + +The first line contains a positive integer **T**, the number of test cases. +**T** test cases follow. + +The first line of each test case contains the number of friends **N**, and +number of friend lists **M**, separated by a space. The second line contains +**x0** and integer parameters **a**, **b**, and **p**, separated by spaces. +You must use these to generate the remaining numbers **x1**, …, **xn-1** +according to this formula: + +> **xi** = (**xi-1** * **a** \+ **b**) mod **p** + +The next M lines of each test case define the friend lists for that test case. +Each line consists of the following integers, separated by spaces: **size**, +**y0**, **a**, **b**. The **size** is the number of friends in the friend +list. **y0** is the index in F of the first friend in the list. You must +generate the remaining friends in the friend list **y1**, …, **yn-1** +according to the following formula: + +> **yi** = (**yi-1** * **a** \+ **b**) mod **n** + +If a friend list contains more than one of a given index, consider it to only +contain that index once. + +## Constraints + +1 ≤ **T** ≤ 20 +2 ≤ **n** ≤ 50 000 +0 ≤ **m** ≤ 1 500 +0 ≤ sum of sizes of all friend lists ≤ 1 000 000 +**Numbers used in generators:** +0 < **a**, **p** < 230 +0 ≤ **b** < 230 + +### Output + +For each of the test cases numbered in order from **1** to **T**, output "Case +#", followed by the case number, followed by ": ", followed by the maximum +possible distance between the two closest user ids that you are still friends +with for that case. + diff --git a/2012/round3/unfriending.out b/2012/round3/unfriending.out new file mode 100644 index 0000000000000000000000000000000000000000..44b76ed393d16678f15d72fee37d53f29ac8c5b5 --- /dev/null +++ b/2012/round3/unfriending.out @@ -0,0 +1,75 @@ +Case #1: 7392 +Case #2: 5130 +Case #3: 15318 +Case #4: 63175 +Case #5: 5814 +Case #6: 4 +Case #7: 3 +Case #8: 3 +Case #9: 6 +Case #10: 4 +Case #11: 3 +Case #12: 3 +Case #13: 6 +Case #14: 6 +Case #15: 6 +Case #16: 3 +Case #17: 3 +Case #18: 3 +Case #19: 3 +Case #20: 4 +Case #21: 3 +Case #22: 4 +Case #23: 5 +Case #24: 3 +Case #25: 7 +Case #26: 3 +Case #27: 10 +Case #28: 8 +Case #29: 3 +Case #30: 6 +Case #31: 9 +Case #32: 3 +Case #33: 3 +Case #34: 4 +Case #35: 6 +Case #36: 4 +Case #37: 3 +Case #38: 4 +Case #39: 3 +Case #40: 4 +Case #41: 4 +Case #42: 6 +Case #43: 4 +Case #44: 3 +Case #45: 6 +Case #46: 3 +Case #47: 6 +Case #48: 4 +Case #49: 7 +Case #50: 4 +Case #51: 3 +Case #52: 4 +Case #53: 5 +Case #54: 3 +Case #55: 3 +Case #56: 3 +Case #57: 4 +Case #58: 4 +Case #59: 6 +Case #60: 6 +Case #61: 3 +Case #62: 5 +Case #63: 3 +Case #64: 4 +Case #65: 4 +Case #66: 4 +Case #67: 3 +Case #68: 10 +Case #69: 6 +Case #70: 4 +Case #71: 6 +Case #72: 4 +Case #73: 5 +Case #74: 4 +Case #75: 3