A natural number is written in each cell of an $8 \times 8$ board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than $32$? [i](N. Chernyatyev)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$

Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.) a) Prove that $(b_n)_{n \ge 0}$ is unbounded. b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.

Find the value of $ a$ for which $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$ is minimized.

The triangle $ABC$ is inscribed in a circle. Two ants crawl out of points $B$ and $C$ at the same time. They crawl along the arc $BC$ towards each other so that the product of the distances from them to point $A$ remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.

Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$. [i]Ray Li[/i]

A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

Let $\Delta ABC$ be an acute triangle and $\omega$ be its circumcircle. Perpendicular from $A$ to $BC$ intersects $BC$ at $D$ and $\omega$ at $K$. Circle through $A$, $D$ and tangent to $BC$ at $D$ intersect $\omega$ at $E$. $AE$ intersects $BC$ at $T$. $TK$ intersects $\omega$ at $S$. Assume, $SD$ intersects $\omega$ at $X$. Prove that $X$ is the reflection of $A$ with respect to the perpendicular bisector of $BC$.

Let $ a > b > c > d $ be positive integers such that \begin{align*} a^2 + ac - c^2 = b^2 + bd - d^2 \end{align*} Prove that $ ab + cd $ is a composite number.

Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

Prove that for any natural number $n$ and real numbers $a$ and $x$ satisfying $a^{n+1} \le x \le 1$ and $0 < a < 1$ it holds that $$\prod_{k=1}^n \left|\frac{x-a^k}{x+a^k}\right| \le \prod_{k=1}^n \frac{1-a^k}{1+a^k}$$

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? $\textbf{(A) } 8 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 14 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 18 $

Let $a_0>0$ be a real number, and let $$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$ Show that $a_{2020}<\frac1{2020}$.

Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$. Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.

Call a day a [i]perfect[/i] day if the sum of the digits of the month plus sum of the digits of the day equals the sum of digits of the year. For example, February $28$th, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$. [i]2018 CCA Math Bonanza Team Round #5[/i]

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$

[u]Round 1[/u] [b]1.1.[/b] A circle has a circumference of $20\pi$ inches. Find its area in terms of $\pi$. [b]1.2.[/b] Let $x, y$ be the solution to the system of equations: $x^2 + y^2 = 10 \,\,\, , \,\,\, x = 3y$. Find $x + y$ where both $x$ and $y$ are greater than zero. [b]1. 3.[/b] Chris deposits $\$ 100$ in a bank account. He then spends $30\%$ of the money in the account on biology books. The next week, he earns some money and the amount of money he has in his account increases by $30 \%$. What percent of his original money does he now have? [u]Round 2[/u] [b]2.1.[/b] The bell rings every $45$ minutes. If the bell rings right before the first class and right after the last class, how many hours are there in a school day with $9$ bells? [b]2.2.[/b] The middle school math team has $9$ members. They want to send $2$ teams to ABMC this year: one full team containing 6 members and one half team containing the other $3$ members. In how many ways can they choose a $6$ person team and a $3$ person team? [b]2.3.[/b] Find the sum: $$1 + (1 - 1)(1^2 + 1 + 1) + (2 - 1)(2^2 + 2 + 1) + (3 - 1)(3^2 + 3 + 1) + ...· + (8 - 1)(8^2 + 8 + 1) + (9 - 1)(9^2 + 9 + 1).$$ [u]Round 3[/u] [b]3.1.[/b] In square $ABHI$, another square $BIEF$ is constructed with diagonal $BI$ (of $ABHI$) as its side. What is the ratio of the area of $BIEF$ to the area of $ABHI$? [b]3.2.[/b] How many ordered pairs of positive integers $(a, b)$ are there such that $a$ and $b$ are both less than $5$, and the value of $ab + 1$ is prime? Recall that, for example, $(2, 3)$ and $(3, 2)$ are considered different ordered pairs. [b]3.3.[/b] Kate Lin drops her right circular ice cream cone with a height of $ 12$ inches and a radius of $5$ inches onto the ground. The cone lands on its side (along the slant height). Determine the distance between the highest point on the cone to the ground. [u]Round 4[/u] [b]4.1.[/b] In a Museum of Fine Mathematics, four sculptures of Euler, Euclid, Fermat, and Allen, one for each statue, are nailed to the ground in a circle. Bob would like to fully paint each statue a single color such that no two adjacent statues are blue. If Bob only has only red and blue paint, in how many ways can he paint the four statues? [b]4.2.[/b] Geo has two circles, one of radius 3 inches and the other of radius $18$ inches, whose centers are $25$ inches apart. Let $A$ be a point on the circle of radius 3 inches, and B be a point on the circle of radius $18$ inches. If segment $\overline{AB}$ is a tangent to both circles that does not intersect the line connecting their centers, find the length of $\overline{AB}$. [b]4.3.[/b] Find the units digit to $2017^{2017!}$. [u]Round 5[/u] [b]5.1.[/b] Given equilateral triangle $\gamma_1$ with vertices $A, B, C$, construct square $ABDE$ such that it does not overlap with $\gamma_1$ (meaning one cannot find a point in common within both of the figures). Similarly, construct square $ACFG$ that does not overlap with $\gamma_1$ and square $CBHI$ that does not overlap with $\gamma_1$. Lines $DE$, $FG$, and $HI$ form an equilateral triangle $\gamma_2$. Find the ratio of the area of $\gamma_2$ to $\gamma_1$ as a fraction. [b]5.2.[/b] A decimal that terminates, like $1/2 = 0.5$ has a repeating block of $0$. A number like $1/3 = 0.\overline{3}$ has a repeating block of length $ 1$ since the fraction bar is only over $ 1$ digit. Similarly, the numbers $0.0\overline{3}$ and $0.6\overline{5}$ have repeating blocks of length $ 1$. Find the number of positive integers $n$ less than $100$ such that $1/n$ has a repeating block of length $ 1$. [b]5.3.[/b] For how many positive integers $n$ between $1$ and $2017$ is the fraction $\frac{n + 6}{2n + 6}$ irreducible? (Irreducibility implies that the greatest common factor of the numerator and the denominator is $1$.) [u]Round 6[/u] [b]6.1.[/b] Consider the binary representations of $2017$, $2017 \cdot 2$, $2017 \cdot 2^2$, $2017 \cdot 2^3$, $... $, $2017 \cdot 2^{100}$. If we take a random digit from any of these binary representations, what is the probability that this digit is a $1$ ? [b]6.2.[/b] Aaron is throwing balls at Carlson’s face. These balls are infinitely small and hit Carlson’s face at only $1$ point. Carlson has a flat, circular face with a radius of $5$ inches. Carlson’s mouth is a circle of radius $ 1$ inch and is concentric with his face. The probability of a ball hitting any point on Carlson’s face is directly proportional to its distance from the center of Carlson’s face (so when you are $2$ times farther away from the center, the probability of hitting that point is $2$ times as large). If Aaron throws one ball, and it is guaranteed to hit Carlson’s face, what is the probability that it lands in Carlson’s mouth? [b]6.3.[/b] The birth years of Atharva, his father, and his paternal grandfather form a geometric sequence. The birth years of Atharva’s sister, their mother, and their grandfather (the same grandfather) form an arithmetic sequence. If Atharva’s sister is $5$ years younger than Atharva and all $5$ people were born less than $200$ years ago (from $2017$), what is Atharva’s mother’s birth year? [u]Round 7[/u] [b]7. 1.[/b] A function $f$ is called an “involution” if $f(f(x)) = x$ for all $x$ in the domain of $f$ and the inverse of $f$ exists. Find the total number of involutions $f$ with domain of integers between $ 1$ and $ 8$ inclusive. [b]7.2.[/b] The function $f(x) = x^3$ is an odd function since each point on $f(x)$ corresponds (through a reflection through the origin) to a point on $f(x)$. For example the point $(-2, -8)$ corresponds to $(2, 8)$. The function $g(x) = x^3 - 3x^2 + 6x - 10$ is a “semi-odd” function, since there is a point $(a, b)$ on the function such that each point on $g(x)$ corresponds to a point on $g(x)$ via a reflection over $(a, b)$. Find $(a, b)$. [b]7.3.[/b] A permutations of the numbers $1, 2, 3, 4, 5$ is an arrangement of the numbers. For example, $12345$ is one arrangement, and $32541$ is another arrangement. Another way to look at permutations is to see each permutation as a function from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$. For example, the permutation $23154$ corresponds to the function f with $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, $f(5) = 4$, and $f(4) = 5$, where $f(x)$ is the $x$-th number of the permutation. But the permutation $23154$ has a cycle of length three since $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, and cycles after $3$ applications of $f$ when regarding a set of $3$ distinct numbers in the domain and range. Similarly the permutation $32541$ has a cycle of length three since $f(5) = 1$, $f(1) = 3$, and $f(3) = 5$. In a permutation of the natural numbers between $ 1$ and $2017$ inclusive, find the expected number of cycles of length $3$. [u]Round 8[/u] [b]8.[/b] Find the number of characters in the problems on the accuracy round test. This does not include spaces and problem numbers (or the periods after problem numbers). For example, “$1$. What’s $5 + 10$?” would contain $11$ characters, namely “$W$,” “$h$,” “$a$,” “$t$,” “$’$,” “$s$,” “$5$,” “$+$,” “$1$,” “$0$,” “?”. If the correct answer is $c$ and your answer is $x$, then your score will be $$\max \left\{ 0, 13 -\left\lceil \frac{|x-c|}{100} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xz+xy+yz=4$, and $x, y, z$ are real numbers

Determine $$ \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|$$ 

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]