Let $S$ and $T$ be non-empty, finite sets of positive integers. We say that $a\in\mathbb{N}$ is \emph{good} for $b\in\mathbb{N}$ if $a\geq\frac{b}{2}+7$. We say that an ordered pair $\left(a,b\right)\in S\times T$ is \emph{satisfiable} if $a$ and $b$ are good for each other. A subset $R$ of $S$ is said to be \emph{unacceptable} if there are less than $\left|R\right|$ elements $b$ of $T$ with the property that there exists $a \in R$ such that $\left(a,b\right)$ is satisfiable. If there are no unacceptable subsets of $S$, and $S$ contains the elements $14$, $20$, $16$, $32$, $23$, and $31$, compute the smallest possible sum of elements of $T$ given that $\left|T\right|\geq20$. [i]Proposed by Tristan Shin[/i]

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.

Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.

Perry the painter wants to paint his floor, but he decides to leave a 1 foot border along the edges. After painting his floor, Perry notices that the area of the painted region is the same as the area of the unpainted region. Perry's floor measures $a$ x $b$ feet, where $a>b$ and both $a$ and $b$ are positive integers. Find all possible ordered pairs $(a, b)$. [i]2016 CCA Math Bonanza Team #2[/i]

Consider a convex quadrilateral $ABCD$. Pairs of its opposite sides are continued until they intersect: $BA$ and $CD$ at the point $P$, $BC$ and $AD$ at the point $Q$. Let $K$ be the intersection point of the exterior bisectors of the angles $A$ and $C$ of the quadrilateral, $L$ be the intersection point of the exterior bisectors of the angles $B$ and $D$ of the quadrilateral, and $M$ be the intersection point of the exterior bisectors of the angles $P$ and $Q$ (the exterior bisector of an angle $X$ is the line passing through X and perpendicular to its ordinary bisector). Prove that the points $K$, $L$ and $M$ lie on a straight line. (S Markelov)

Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that $$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$

Let $n\in\mathbb{Z}^+$. Arrange $n$ students $A_1,A_2,...,A_n$ on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is $m\geq n$. A configuration is called [i]balance[/i] if for an arbitrary student $A_i$, there will always be a regular polygon taking $A_i$ as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies. a) Given $n$, find the least $m$ such that we can create a balance configuration. b) In a [i]move[/i], a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if $n$ is the product of at most $2$ prime numbers and $m$ satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum $$p_1^4+p_2^4+\dots +p_{2021}^4$$ is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$. $\textit{Stefan Bălăucă}$

Let $A{}$ and $B{}$ be $3\times 3{}$ matrices with complex entries, satisfying $A^2=B^2=O_3$. Prove that if $A{}$ and $B{}$ commute, then $AB=O_3$. Is the converse true?

The integers $1, 2,. . . , n$ are arranged in order so that each value is strictly larger than all values above or is strictly less than all values previous. In how many ways can this be done?

What is the hybridization of the carbon atom in a carboxyl group? $ \textbf{(A) }sp \qquad\textbf{(B) } sp^2\qquad\textbf{(C) } sp^3\qquad\textbf{(D) } dsp^3\qquad $

Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$.

Compute $$\int\frac{(\sin x+\cos x)(4-2\sin2x-\sin^22x)e^x}{\sin^32x}dx$$ where $x\in\left(0,\frac\pi2\right)$. [i]Proposed by Mihály Bencze[/i]

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

On a board are drawn the points $A,B,C,D$. Yetti constructs the points $A^\prime,B^\prime,C^\prime,D^\prime$ in the following way: $A^\prime$ is the symmetric of $A$ with respect to $B$, $B^\prime$ is the symmetric of $B$ wrt $C$, $C^\prime$ is the symmetric of $C$ wrt $D$ and $D^\prime$ is the symmetric of $D$ wrt $A$. Suppose that Armpist erases the points $A,B,C,D$. Can Yetti rebuild them? $\star \, \, \star \, \, \star$ [b]Note.[/b] [i]Any similarity to real persons is purely accidental.[/i]

[b]p1.[/b] Denote $S_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}$. What is $144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})$? [b]p2.[/b] Let $A,B,C$ be three collinear points, with $AB = 4$, $BC = 8$, and $AC = 12$. Draw circles with diameters $AB$, $BC$, and $AC$. Find the radius of the two identical circles that will lie tangent to all three circles. [b]p3.[/b] Let $s(i)$ denote the number of $1$’s in the binary representation of $i$. What is $$\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right) \,\, mod \,\,629 ?$$ [b]p4.[/b] Parallelogram $ABCD$ has an area of $S$. Let $k = 42$. $E$ is drawn on AB such that $AE =\frac{AB}{k}$ . $F$ is drawn on $CD$ such that $CF = \frac{CD}{k}$ . $G$ is drawn on $BC$ such that $BG = \frac{BC}{k}$ . $H$ is drawn on $AD$ such that $DH = \frac{AD}{k}$ . Line $CE$ intersects $BH$ at $M$, and $DG$ at $N$. Line $AF$ intersects $DG$ at $P$, and $BH$ at $Q$. If $S_1$ is the area of quadrilateral $MNPQ$, find $\frac{S_1}{S}$. [b]p5.[/b] Let $\phi$ be the Euler totient function. What is the sum of all $n$ for which $\frac{n}{\phi(n)}$ is maximal for $1 \le n \le 500$? [b]p6.[/b] Link starts at the top left corner of an $12 \times 12$ grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly $6$ turns, in how many ways can he reach his destination? PS. You had better use hide for answers.

Let \[ f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .\] Prove that \[ \limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .\] [i]P. Erdos[/i]

Inscribe in the triangle $ABC$ a triangle with minimum perimeter.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true: $$f(x+y)^2=f(x+y)+f(x)+f(y)$$ [i]D. Zmiaikou[/i]

Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?

Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let $$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$ Prove that $f_n$ satisfy the following recursion: $$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$

Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.

Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.