The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying \[ f(x)=\begin{cases} 3x+1,& \mbox{as }x\mbox{ is odd}\\ x/2, & \mbox{as }x\mbox{ is even.}\\ \end{cases} \] In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying \[f^{40}(x)> 2012x.\]

For any set $S$ of integers, let $f(S)$ denote the number of integers $k$ with $0 \le k < 2019$ such that there exist $s_1, s_2 \in S$ satisfying $s_1 - s_2 = k$. For any positive integer $m$, let $x_m$ be the minimum possible value of $f(S_1) + \dots + f(S_m)$ where $S_1, \dots, S_m$ are nonempty sets partitioning the positive integers. Let $M$ be the minimum of $x_1, x_2, \dots$, and let $N$ be the number of positive integers $m$ such that $x_m = M$. Compute $100M + N$. [i]Proposed by Ankan Bhattacharya[/i]

Let $q$ be a monic polynomial with integer coefficients. Prove that there exists a constant $C$ depending only on polynomial $q$ such that for an arbitrary prime number $p$ and an arbitrary positive integer $N \leq p$ the congruence $n! \equiv q(n) \pmod p$ has at most $CN^\frac {2}{3}$ solutions among any $N$ consecutive integers.

Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.

On a chessboard $5 \times 9$ squares, the following game is played. Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules: $\bullet$ Each frog may be moved one square up, down, left, or right; $\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa; $\bullet$ At the end of each turn, no square can contain two or more frogs. The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

Show that $r = 2$ is the largest real number $r$ which satisfies the following condition: If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities \[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\] for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$

For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$ Find all possible values of $a+b+c+d$

In the symmetric group \(S_n\ (n \geq 3)\), let \(G_{a,b}\) be the subgroup generated by the 2-cycle \((a\ b)\) and the n-cycle \((1\ 2\ \cdots\ n)\). Find the index \(\left|S_n : G_{a,b}\right|\).

A sequence $\{a_n\} $ satisfies $a_1$ is a positive integer and $a_{n+1}$ is the largest odd integer that divides $2^n-1+a_n$ for all $n\geqslant 1$. Given a positive integer $r$ which is greater than $1$. Is it possible that there exists infinitely many pairs of ordered positive integers $(m,n)$ for which $m>n$ and $a_m = ra_n$? In other words, if you successfully find [b]an[/b] $a_1$ that yields infinitely many pairs of $(m,n)$ which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible $a_1$. @below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.

Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties: $\: \bullet \: \gcd(a, b, c) = G$. $\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$. $\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers. $\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.

Let $N$ be the number of sequences of positive integers greater than $ 1$ where the product of all of the terms of the sequence is $12^{64}$. If $N$ can be expressed as $a(2^b)$ ), where $a$ is an odd positive integer, determine $b$.

Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

Let $S$, $K$, $I$, $B$, $D$, $Y$ be distinct integers from $0$ to $9,$ inclusive. Given that they follow this equation: $$\begin{array}{rrrrr} & S & K & I & B \\ - & I & D & I & D \\ \hline & & & D & Y \end{array}$$find the maximum value of $\overline{SKIBIDI}$.

The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

In triangle $ABC$, the incircle touches $BC$ in $D$, $CA$ in $E$ and $AB$ in $F$. The bisector of $\angle BAC$ intersects $BC$ in $G$. The lines $BE$ and $CF$ intersect in $J$. The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$. Prove that $\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$

Let $p_1, p_2, ..., p_k$ be $k$ different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence: $a_1 < a_2 < ... < a_n < ...$ Prove that, for every number $c$, there exists $n$ such that $a_{n+1} -a_n > c$.

Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.

Katie Ledecky and Michael Phelps each participate in $7$ swimming events in the Olympics (and there is no event that they both participate in). Ledecky receives $g_L$ gold, $s_L$ silver, and $b_L$ bronze medals, and Phelps receives $g_P$ gold, $s_P$ silver, and $b_P$ bronze medals. Ledecky notices that she performed objectively better than Phelps: for all positive real numbers $w_b<w_s<w_g$, we have $$w_gg_l+w_ss_L+w_bb_L>w_gg_P+w_ss_P+w_bb_P.$$ Compute the number of possible $6$-tuples $(g_L,s_L,b_L,g_P,s_P,b_P).$

There are $8$ different quadratic trinomials written on the board, among them there are no two that add up to a zero polynomial. It turns out that if we choose any two trinomials $g_1(x), g_2(X)$ from the board, then the remaining $6$ trinomials can be denoted as $g_3(x),g_4(x),\dots,g_8(x)$ so that all four polynomials $g_1(x)+g_2(x),g_3(x)+g_4(x),g_5(x)+g_6(x)$ and $g_7(x)+g_8(x)$ have a common root. Do all trinomials on the board necessarily have a common root? [i]Proposed by S. Berlov[/i]