problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$ , different from $A$ and $B$ . The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$ . The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$ . If the perimeter of the triangle $PEQ$ is $24$ , find the length of the side $PQ$ | 8 | 0.875 |
Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\text{cm}$ . Another line intersects the circles at $E,F$ , respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$ . Find the radius $r$ . | 13 | 0.25 |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$ , where $m$ , $n$ are relatively prime integers. Find $m + n$ . | 2224 | 0.75 |
The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$ . The circle $s$ with diameter $AH$ ( $H$ is the orthocenter of $ABC$ ) meets $\omega$ for the second time at point $P$ . Restore the triangle $ABC$ if the points $A$ , $P$ , $W$ are given. | ABC | 0.875 |
Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ . What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$ ? | 15 | 0.375 |
Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$ $:$ { $1,2,...n$ } $\rightarrow$ { $1,2,...,n$ } and a binary function $C:$ { $1,2,...,n$ } $\rightarrow$ { $0,1$ } "revengeful" if it satisfies the two following conditions: $1)$ For every $i$ $\in$ { $1,2,...,n$ }, there exist $j$ $\in$ $S_{i}=$ { $i, \pi(i),\pi(\pi(i)),...$ } such that $C(j)=1$ . $2)$ If $C(k)=1$ , then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$ , where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$ , for every positive integer $t$ .
Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$ .
Determine $\frac{V}{P}$ . | n! | 0.75 |
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$ , $x_2$ , $\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .
*Proposed by Le Duc Minh* | 64 | 0.75 |
An triangle with coordinates $(x_1,y_1)$ , $(x_2, y_2)$ , $(x_3,y_3)$ has centroid at $(1,1)$ . The ratio between the lengths of the sides of the triangle is $3:4:5$ . Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
the area of the triangle can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*2021 CCA Math Bonanza Individual Round #11* | 107 | 0.25 |
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won? | 11 | 0.375 |
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
**1.**Each plane contains at least $ 4$ of them
**2.**No four points are collinear. | 6 | 0.125 |
For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$ entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$ .
| 1 | 0.75 |
(1) For $ a>0,\ b\geq 0$ , Compare
$ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$ .
(2) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$ . | 1 | 0.5 |
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
| x = 1972 | 0.625 |
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$ , $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$ . A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$ , and $G_0\in \triangle A_1A_2A_3$ ; what is $s$ ? | \frac{180}{71} | 0.875 |
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$ .
(Gerhard J. Woeginger) | 13 | 0.25 |
Find the sum of all distinct possible values of $x^2-4x+100$ , where $x$ is an integer between 1 and 100, inclusive.
*Proposed by Robin Park* | 328053 | 0.25 |
Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$ . Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$ . | 1 | 0.875 |
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$ . Show that $f(8) > 2022f(0)$ .
*Proposed by Ethan Tan* | f(8) > 2022 f(0) | 0.875 |
What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
*Ray Li* | 36 | 0.875 |
Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$ . Let $N$ be the reflection of $M$ in $BC$ . The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$ . Determine the value of $\angle AKC$ .
(A.Blinkov) | 90^\circ | 0.875 |
At least $ n - 1$ numbers are removed from the set $\{1, 2, \ldots, 2n - 1\}$ according to the following rules:
(i) If $ a$ is removed, so is $ 2a$ ;
(ii) If $ a$ and $ b$ are removed, so is $ a \plus{} b$ .
Find the way of removing numbers such that the sum of the remaining numbers is maximum possible. | n^2 | 0.5 |
The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$ . Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$ , we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$ . | \frac{n+1}{n-1} | 0.125 |
A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$ . Let $A'$ be a rotation of $A$ in the plane by $60^\circ$ . The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$ . Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$ . What is the greatest possible value of the number $x$ ? | 150^\circ | 0.75 |
Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs $(i, j)$ , $1\leq i,j\leq 2022$ , satisfying $$ a_i^2+a_j\ge \frac 1{2021}. $$ | 2021 \times 2022 | 0.125 |
5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$ . (We include 1 in the set $S$ .) If $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$ , find $a+b$ . (Here $\varphi$ denotes Euler's totient function.) | 1537 | 0.75 |
The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let $ S_k(n)$ denote the elementary expression on $ k$ different letters of order $ n;$ for example $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$ There are four terms in $ S_4(3).$ How many terms are there in $ S_{9891}(1989)?$ (Assume that we have $ 9891$ different letters.) | \binom{9891}{1989} | 0.875 |
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$ , let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$ , we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$ , the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need? | 2 | 0.25 |
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face? | 6 | 0.875 |
Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$ . | \frac{1}{4} | 0.5 |
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction). | 168 | 0.5 |
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.) | 93 | 0.625 |
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$ |S| +\ min(S) \cdot \max (S) = 0. $$ | 335 | 0.125 |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers. | 95 | 0.125 |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$ , let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$ , let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$ , and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$ , $ P(1) \equal{} y_1$ , $ P(2) \equal{} y_2$ , and $ P(3) \equal{} y_3$ . What is the expected value of $ P(4)$ ? | 107 | 0.625 |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$ .
What is the area of triangle $ABC$ ? | 200 | 0.875 |
To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$ . Solve the equation $x*36 = 216$ . | 7776 | 0.125 |
How many ways are there to insert $+$ 's between the digits of $111111111111111$ (fifteen $1$ 's) so that the result will be a multiple of $30$ ? | 2002 | 0.5 |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For example, if she cuts the $1111$ th link out of her chain first, then she will have $3$ chains, of lengths $1110$ , $1$ , and $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from $1$ to $2018$ links using some combination of her chains?
*2018 CCA Math Bonanza Individual Round #10* | 10 | 0.5 |
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$ | 9 | 0.75 |
In an acute scalene triangle $ABC$ , points $D,E,F$ lie on sides $BC, CA, AB$ , respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes $AD, BE, CF$ meet at orthocenter $H$ . Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$ . Lines $DP$ and $QH$ intersect at point $R$ . Compute $HQ/HR$ .
*Proposed by Zuming Feng* | 1 | 0.375 |
For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$ | 19 | 0.625 |
Find a costant $C$ , such that $$ \frac{S}{ab+bc+ca}\le C $$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.) | \frac{1}{4\sqrt{3}} | 0.5 |
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her? | 5 | 0.875 |
Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$ | 1 | 0.875 |
Evaluate $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$ , where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$ . | \frac{\pi}{2} | 0.75 |
Construct the $ \triangle ABC$ , given $ h_a$ , $ h_b$ (the altitudes from $ A$ and $ B$ ) and $ m_a$ , the median from the vertex $ A$ . | \triangle ABC | 0.625 |
Let $a_1=24$ and form the sequence $a_n$ , $n\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ? | 88 | 0.375 |
Find the number of ordered quadruples of positive integers $(a,b,c, d)$ such that $ab + cd = 10$ . | 58 | 0.75 |
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \cdot B$ .
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$ .
| 143 | 0.5 |
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique **alternating sum** is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$ . | 448 | 0.625 |
A four-digit positive integer is called *virtual* if it has the form $\overline{abab}$ , where $a$ and $b$ are digits and $a \neq 0$ . For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$ , for some positive integer $n$ . | 8282 | 0.75 |
Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$ . Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$ , what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$ ?
*2018 CCA Math Bonanza Lightning Round #4.1* | 1009k | 0.875 |
On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$ . He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$ . Find the difference between largest and smallest values Donald can achieve. | 0 | 0.5 |
Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by
\[\tau(f)(x)=f(x+1).\]
Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$ . Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$ . | g = x^{p-1} | 0.25 |
The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$ . Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$ ? Point A on the board located above and to the left of point $B$ . | O(0, 0) | 0.375 |
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$ | n = 3 | 0.125 |
Pick out three numbers from $0,1,\cdots,9$ , their sum is an even number and not less than $10$ . We have________different ways to pick numbers. | 51 | 0.375 |
If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$ , what is the probability that $|a|+|b|<1$ ? | \frac{1}{2} | 0.875 |
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$ , compute $P(1)$ .
| 100 | 0.75 |
Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes. | 175 | 0.875 |
Let $n\in \mathbb{Z}_{> 0}$ . The set $S$ contains all positive integers written in decimal form that simultaneously satisfy the following conditions:
[list=1][*] each element of $S$ has exactly $n$ digits;
[*] each element of $S$ is divisible by $3$ ;
[*] each element of $S$ has all its digits from the set $\{3,5,7,9\}$ [/list]
Find $\mid S\mid$ | \frac{4^n + 2}{3} | 0.875 |
Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$ . Find all possible values of $x+y$ .
| 0, 3, 4, 5, 8 | 0.875 |
Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty. | K = 1011 | 0.25 |
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$ , such that $a_j = a_i + 1$ and $a_k = a_j + 1$ . Considering all such sequences $A$ , find the greatest value of $m$ . | 667^3 | 0.25 |
How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ? | 120 | 0.25 |
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers. | n = 3 | 0.375 |
In triangle $ABC$ , let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$ . Find $OH$ .
*Proposed by Kevin You* | 10 | 0.125 |
A positive integer is called *oneic* if it consists of only $1$ 's. For example, the smallest three oneic numbers are $1$ , $11$ , and $111$ . Find the number of $1$ 's in the smallest oneic number that is divisible by $63$ . | 18 | 0.875 |
Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$ , then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$ . | n | 0.125 |
Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | 33 | 0.75 |
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists. | 5 | 0.875 |
Let $f(x)$ be a function such that $f(1) = 1234$ , $f(2)=1800$ , and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$ . Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
*2022 CCA Math Bonanza Tiebreaker Round #4* | 8092 | 0.625 |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer. | 725 | 0.875 |
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors. | 130 | 0.75 |
Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \] | (6, 2, 10) | 0.75 |
Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team? | \frac{8}{17} | 0.375 |
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$ , it is true that at least one of the following numbers: $$ a, b,\frac{5}{a^2}+\frac{6}{b^3} $$ is less than or equal to $k$ . | 2 | 0.75 |
Positive sequences $\{a_n\},\{b_n\}$ satisfy: $a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$ .
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$ ,where $m$ is a given positive integer. | 5 | 0.625 |
Leo the fox has a $5$ by $5$ checkerboard grid with alternating red and black squares. He fills in the
grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares
(sharing a side) and each number is used exactly once. He then computes the sum of the numbers in
the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo
can obtain. | 169 | 0.5 |
Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ . | 907200 | 0.5 |
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals. | 49 | 0.625 |
Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost? | 7 | 0.625 |
Let $ABCD$ be a square of side length $1$ , and let $E$ and $F$ be points on $BC$ and $DC$ such that $\angle{EAF}=30^\circ$ and $CE=CF$ . Determine the length of $BD$ .
*2015 CCA Math Bonanza Lightning Round #4.2* | \sqrt{2} | 0.75 |
Alex starts with a rooted tree with one vertex (the root). For a vertex $v$ , let the size of the subtree of $v$ be $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$ ). The expected value of $S(v_1)$ is of the form $\tfrac{m}{n}$ for relatively prime positive integers $m, n$ . Find $m+n$ .**Note:** In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself).
*Proposed by Yang Liu* | 9901 | 0.375 |
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign " $+$ " or a multiplication sign " $\times$ " between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$ , with value 13.) Each positive digit is equally likely, each arithmetic sign (" $+$ " or " $\times$ ") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
| 50 | 0.875 |
Let $B_n$ be the set of all sequences of length $n$ , consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$ \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1). $$ . Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $ .
. | n \cdot 4^{n-1} | 0.125 |
In the quadrilateral $ABCD$ , we have $\measuredangle BAD = 100^{\circ}$ , $\measuredangle BCD = 130^{\circ}$ , and $AB=AD=1$ centimeter. Find the length of diagonal $AC$ . | 1 \text{ cm} | 0.25 |
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?
Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$ . | 13 | 0.25 |
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$ . How many progressions does Omar's list have? | 6 | 0.5 |
Let $ABC$ be a triangle with $|AB|=|AC|=26$ , $|BC|=20$ . The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$ , respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$ . | \frac{65}{12} | 0.25 |
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by *Nikola Velov, Macedonia* | 100 | 0.875 |
Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$ . | 45^\circ | 0.75 |
Let $p$ be a permutation of the set $S_n = \{1, 2, \dots, n\}$ . An element $j \in S_n$ is called a fixed point of $p$ if $p(j) = j$ . Let $f_n$ be the number of permutations having no fixed points, and $g_n$ be the number with exactly one fixed point. Show that $|f_n - g_n| = 1$ . | |f_n - g_n| = 1 | 0.625 |
In triangle $ABC$ , side $AB$ has length $10$ , and the $A$ - and $B$ -medians have length $9$ and $12$ , respectively. Compute the area of the triangle.
*Proposed by Yannick Yao* | 72 | 0.875 |
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals $5$ red chips, $3$ red tile are equal to $ 8$ black tiles, and a black tile is worth $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to $560$ and there are no more than five tiles of the same color. | 3 | 0.75 |
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$ .
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated. :) | \frac{1}{8} | 0.625 |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$ , and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$ ? | 578 | 0.375 |
Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ( $1 \times 1$ squares) are *neighbors* if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.
An integer from 1 to 4 is written inside each square according to the following rules:
- If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
- If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
- If a cell has a 4 written on it, then all of its neighbors contain a 1.
Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid? | \frac{5n^2}{2} | 0.75 |
In $\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$ . Find $BP$ .
*Proposed by Eugene Chen* | 40 | 0.5 |
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$ . | 865 | 0.625 |
Subsets and Splits