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$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 | 0.75 |
Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$ . | 48 | 0.75 |
Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). | 89 | 0.375 |
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
*Mircea Becheanu*. | p | 0.75 |
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$ . If $x = 0$ is a root of $f(x) = 0$ , what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$ ? | 401 | 0.625 |
For each positive integer $ k$ , find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$ ,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$ , and
(3) $ A_{1}A_{2}\ldots A_{k}\ne 0$ . | n_k = 2^k | 0.25 |
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$ . He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have. | 1120 | 0.5 |
In right triangle $ ABC$ with right angle at $ C$ , $ \angle BAC < 45$ degrees and $ AB \equal{} 4$ . Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$ . The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$ , where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$ . | 7 | 0.25 |
Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$ . Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$ . | 21 | 0.75 |
Given that the base- $17$ integer $\overline{8323a02421_{17}}$ (where a is a base- $17$ digit) is divisible by $\overline{16_{10}}$ , find $a$ . Express your answer in base $10$ .
*Proposed by Jonathan Liu* | 7 | 0.375 |
Wendy randomly chooses a positive integer less than or equal to $2020$ . The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 107 | 0.25 |
Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $ , $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.
| 2016 | 0.25 |
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located. | \frac{\pi}{4} | 0.875 |
A circle of radius $4$ is inscribed in a triangle $ABC$ . We call $D$ the touchpoint between the circle and side BC. Let $CD =8$ , $DB= 10$ . What is the length of the sides $AB$ and $AC$ ? | 12.5 | 0.125 |
The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$ . Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$ . | 41 | 0.75 |
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$ . | 24 | 0.625 |
Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares. | 4 | 0.75 |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ . | 90 | 0.25 |
Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$ | 1004 | 0.75 |
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$ . It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$ .
*Proposed by Evan Chen* | 45 | 0.125 |
Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained. | 0 | 0.5 |
Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$ th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.
*Proposed by Connor Gordon* | \frac{25}{101} | 0.625 |
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen? | 1008 | 0.125 |
Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there? | 4544 | 0.125 |
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .
| 440 | 0.75 |
Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$ , $BI=\sqrt{5}$ , $CI=\sqrt{10}$ and the inradius is $1$ . Let $A'$ be the reflection of $I$ across $BC$ , $B'$ the reflection across $AC$ , and $C'$ the reflection across $AB$ . Compute the area of triangle $A'B'C'$ . | \frac{24}{5} | 0.625 |
The sequences $(a_{n})$ , $(b_{n})$ are defined by $a_{1} = \alpha$ , $b_{1} = \beta$ , $a_{n+1} = \alpha a_{n} - \beta b_{n}$ , $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$ ? | 1999 | 0.25 |
Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions. | k = 4 | 0.375 |
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$ . Determine the length of the hypotenuse. | 994010 | 0.625 |
Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$ , where $n$ is an integer greater than $1$ . Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.) | k(n) = n + 1 | 0.875 |
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | 5 | 0.625 |
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .
*2021 CCA Math Bonanza Lightning Round #2.4* | 14 | 0.75 |
A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$ . A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$ . A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation. | \frac{2}{15} | 0.25 |
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$ , where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$ .
*Proposed by firebolt360* | 360 | 0.125 |
Let $\alpha$ and $\beta$ be positive integers such that $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$ . | 23 | 0.5 |
The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$ subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$ , then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$ . | 3 | 0.25 |
Call a $ 3$ -digit number *geometric* if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. | 840 | 0.5 |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine $ n$ . | 2002 | 0.875 |
In triangle $ABC$ , find the smallest possible value of $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$ | \frac{8\sqrt{3}}{9} | 0.625 |
Alice is counting up by fives, starting with the number $3$ . Meanwhile, Bob is counting down by fours, starting with the number $2021$ . How many numbers between $3$ and $2021$ , inclusive, are counted by both Alice and Bob? | 101 | 0.5 |
Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$ , compute $\frac{1}{c}$ .
*2021 CCA Math Bonanza Lightning Round #2.2* | 16 | 0.875 |
What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$ ?
*2016 CCA Math Bonanza Lightning #1.1* | 7 | 0.875 |
Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$ . Which one of the numbers $x^y$ , $y^x$ is bigger ? | x^y = y^x | 0.875 |
If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ? | 2011 | 0.75 |
Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\] | (2014, 2014, 2014) | 0.125 |
$x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ . | 198 | 0.625 |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$ .
*Ray Li* | 40320 | 0.625 |
Let $r$ be a positive integer. Show that if a graph $G$ has no cycles of length at most $2r$ , then it has at most $|V|^{2016}$ cycles of length exactly $2016r$ , where $|V|$ denotes the number of vertices in the graph $G$ . | |V|^{2016} | 0.625 |
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$ th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days. | 3 | 0.5 |
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$ , there exists an unique positive integer $k$ , satisfying $f^k(n)\leq n+k+1$ . | f(n) = n + 2 | 0.875 |
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours? | 3 | 0.625 |
Determine all positive integers $n$ such that $$ n\cdot 2^{n-1}+1 $$ is a perfect square. | 5 | 0.75 |
Let $f(x) = 2^x + 3^x$ . For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$ ? | 11 | 0.75 |
The altitudes of the triangle ${ABC}$ meet in the point ${H}$ . You know that ${AB = CH}$ . Determine the value of the angle $\widehat{BCA}$ .
| 45^\circ | 0.75 |
A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.
How many positive integers $n$ are there with $2018\leq n \leq 3018$ , such that there exists a collection of $n$ squares that is tri-connected? | 501 | 0.625 |
Let $T=TNFTPP$ . Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$ , $CD = 5$ , and $BD = 6$ . Determine the area of quadrilateral $ACDB$ .
[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=10$</details>. | 44 | 0.625 |
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime. | 17 | 0.25 |
Let $a_1$ , $a_2$ , $a_3$ , $a_4$ , $a_5$ be real numbers satisfying
\begin{align*}
2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4
2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3
2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2
2a_4+a_5 &= 6 + a_1
\end{align*}
Compute $a_1+a_2+a_3+a_4+a_5$ .
*Proposed by Evan Chen* | 2 | 0.125 |
For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$ . Find the smallest $n$ for which
\[ P(P(P(n))) > 10^{12} \] | 6 | 0.375 |
Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$ . Here we make the convention $\sqrt[1]{abc}=abc$ . | 3 | 0.375 |
Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$ ? | 5 | 0.875 |
Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that
- each member in $S$ is a positive integer not exceeding $2002$ ,
- if $a,b\in S$ (not necessarily different), then $ab\not\in S$ .
| 1958 | 0.875 |
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$ . | 1 | 0.5 |
Let $\triangle ABC$ be a triangle with $AB < AC$ . Let the angle bisector of $\angle BAC$ meet $BC$ at $D$ , and let $M$ be the midpoint of $BC$ . Let $P$ be the foot of the perpendicular from $B$ to $AD$ . $Q$ the intersection of $BP$ and $AM$ . Show that : $(DQ) // (AB) $ . | DQ \parallel AB | 0.375 |
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$ ; he notices that the expected value of the score of the resulting grid is equal to $k$ , too! Given that $k > 0.9999$ , find the minimum possible value of $n$ .
*Proposed by Andrew Wu* | 51 | 0.875 |
Determine all positive integers $n$ for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution. | n=1 | 0.75 |
The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots.
\]
Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$ . Find $m+n$ . | 61 | 0.5 |
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer. | 1 \text{ and } 27 | 0.5 |
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$ , but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$ ). Find the maximum number of contestants. | 64 | 0.25 |
Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$ ? | (0, A^2) | 0.625 |
We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions. | 4n^2 - 4n + 2 | 0.25 |
There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$ . Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there?
*Ray Li* | 501 | 0.875 |
More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions. | n = 12 | 0.25 |
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime? | 25 | 0.75 |
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once? | \frac{1}{4} | 0.75 |
A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$ .
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$ . | 7 | 0.125 |
Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$ ? | 333 | 0.875 |
In triangle $ABC$ , angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$ , and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$ | 291 | 0.625 |
Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$ , such that each triangle in $T$ has a side that is not a side of any other triangle in $T$ ? | n | 0.875 |
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working? | 9 | 0.375 |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? | 36 | 0.5 |
Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$ , $3 \nmid q$ , $p^3 = r^3 - q^2$ . | p = 7 | 0.375 |
The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$ , where $m$ , $n$ , and $p$ are integers. Find $m+n+p$ . | 348 | 0.5 |
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$ | -(x - 1)^2 e^x + C | 0.125 |
Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$ . Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$ . If the lines $AD$ and $OE$ meet at $F$ , find $|AF|/|FD|$ . | \frac{1}{2} | 0.625 |
Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum. | \frac{3}{4} | 0.875 |
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
*Proposed by Nathan Xiong* | 16 | 0.75 |
Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | 4 | 0.625 |
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data? | 990 | 0.75 |
Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$ . If $\angle ACB = 40^o$ , $AD = 7$ , $CD = 24$ , $CE = 15$ , and $BE = 20$ , what is the measure of $\angle ABC $ in,degrees? | 70^\circ | 0.875 |
Let $n$ be a natural number divisible by $3$ . We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ( $m > 1$ ), the number of black squares is not more than white squares. Find the maximum number of black squares. | \frac{4n^2}{9} | 0.5 |
Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ *Adrian Zanoschi* | (0, 6) | 0.375 |
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$ . If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$ . For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]
As $n$ approaches infinity, what value does $n a_n$ approach? | \frac{34}{19} | 0.375 |
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ( $ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$ . For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$ . Find the smallest positive integer $ t$ , such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$ . | t = 2 | 0.125 |
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$ , $a_n=1$ , and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ( $F_0=0$ , $F_1=1$ , and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$ ). This representation is said to be $\textit{minimal}$ if it has fewer 1’s than any other Fibonacci representation of $k$ . Find the smallest positive integer that has eight ones in its minimal Fibonacci representation. | 1596 | 0.375 |
Find all positive integers $n>1$ such that
\[\tau(n)+\phi(n)=n+1\]
Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$ , and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$ .
*Raja Oktovin, Pekanbaru* | n = 4 | 0.625 |
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws. | \frac{55}{64} | 0.875 |
Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by
\[ T_n(w) = w^{w^{\cdots^{w}}},\]
with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $ T_2(3) = 3^3 = 27.$ Find the smallest tower of $ 3's$ that exceeds the tower of $ 1989$ $ 2's.$ In other words, find the smallest value of $ n$ such that $ T_n(3) > T_{1989}(2).$ Justify your answer. | 1988 | 0.375 |
For two sets $A, B$ , define the operation $$ A \otimes B = \{x \mid x=ab+a+b, a \in A, b \in B\}. $$ Set $A=\{0, 2, 4, \cdots, 18\}$ and $B=\{98, 99, 100\}$ . Compute the sum of all the elements in $A \otimes B$ .
*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 7)* | 29970 | 0.5 |
On semicircle, with diameter $|AB|=d$ , are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$ | 8 | 0.875 |
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