Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.

For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer. [i]Proposed by Evan Chen[/i]

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \] [b]OR[/b] Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $ \$56$. The absolute difference between the amounts Ashley and Betty had to spend was $ \$19$. The absolute difference between the amounts Betty and Carlos had was $ \$7$, between Carlos and Dick was $ \$5$, between Dick and Elgin was $ \$4$, and between Elgin and Ashley was $ \$11$. How much did Elgin have? $ \textbf{(A)}\ \$6\qquad \textbf{(B)}\ \$7\qquad \textbf{(C)}\ \$8\qquad \textbf{(D)}\ \$9\qquad \textbf{(E)}\ \$10$

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\,\triangle ABC$ touches side $BC$. The circumcircle of $\triangle BDE$ meets line $AB$ again at point $K$, the circumcircle of $\triangle CDF$ meets line $AC$ again at point $L$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\,\triangle ABC$. [i]Proposed by Luu Dong, Vietnam[/i]

Consider a string of $n$ 7's, $7777\cdots77$, into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight 7's in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value 7000?

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.

Let $ABC$ be an acute triangle. Let $D$ be a point inside side $BC$. Let $E$ be the foot from $D$ to $AC$, and let $F$ be a point on $AB$ so that $FE\perp AB$. It is given that the lines $AD, BE, CF$ concur. $M_A, M_B, M_C$ are the midpoints of sides $BC, AC, AB$ respectively, and $O$ is the circumcenter of $ABC$. Moreover, we define $P=EF\cap M_AM_B, S=DE\cap M_AM_C$. Prove that $O, P, S$ are collinear.

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.

Consider $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Prove that $ g\in\Lambda $ implies $ g'\in\Lambda . $

Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .

If $ \displaystyle \sum_{n \equal{} 0}^{\infty} \cos^{2n} \theta \equal{} 5$, what is the value of $ \cos{2\theta}$? $ \textbf{(A)}\ \frac15 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac {\sqrt5}{5}\qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac45$

Determine the sum of the two largest prime factors of the integer $89! + 90!.$

a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$. b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.

The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy: \[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

A [i]word[/i] is a string of the symbols $a, b$ which can be formed by repeated application of the following: (1) $ab$ is a word; (2) if $X$ and $Y$ are words, then so is $XY$; (3) if $X$ is a word, then so is $aXb$. How many words have $12$ letters?

Find the number of ways to completely cover a $2 \times 10$ rectangular grid of unit squares with $2 \times 1$ rectangles $R$ and $\sqrt{2}$ - $\sqrt{2}$ - $2$ triangles $T$ such that the following all hold: [list] [*] a placement of $R$ must have all of its sides parallel to the grid lines, [*] a placement of $T$ must have its longest side parallel to a grid line, [*] the tiles are non-overlapping, and [*] no tile extends outside the boundary of the grid. [/list] (The figure below shows an example of such a tiling; consider tilings that differ by reflections to be distinct.) [asy] unitsize(1cm); fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey); draw((0,0)--(10,0)--(10,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((2,1)--(4,1)); draw((5,0)--(5,2)); draw((7,0)--(7,2)); draw((5,1)--(7,1)); draw((8,0)--(8,2)); draw((8,0)--(10,2)); draw((8,2)--(10,0)); [/asy]

$300$ couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly $2/3$ of the men say their partner always tells the truth and the remaining $1/3$ say their partner always lies. Exactly $2/3$ of the women say their partner is the same type as themselves and the remaining $1/3$ say their partner is different. Find $a$, the maximum possible number of people who tell the truth, and $b$, the minimum possible number of people who tell the truth. Express your answer as $(a,b)$.

Find all possible pairs of integers $ a$ and $ b$ such that $ab = 160 + 90 (a,b)$, where $(a, b)$ is the greatest common divisor of $ a$ and $ b$.