We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

We have a large supply of black, white, red and green hats. And we want to give $8$ of these hats to $8$ students that are sitting around a round table. Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will notchange their places, and that hats of the same color are identical) a) Each hat to be used must be either red or green. b) Exactly two hats of each color are to be used c) Exactly two hats of each color are to be used, and every two hats of the same color are to be given to two adjacent students. d) Exactly two hats of each color are to be used, and no two hats of the same color are to be given to two adjacent students. e) There are no restrictions on the number of hats of each color that are to be used, but no two hats of the same color are to be given to two adjacent students.

The value of $ 10^{\log_{10}7}$ is: $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \log_{10} 7 \qquad\textbf{(E)}\ \log_7 10$

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?

Let $a_n$ and $b_n$ to be two sequences defined as below: $i)$ $a_1 = 1$ $ii)$ $a_n + b_n = 6n - 1$ $iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Determine $a_{2009}$.

Let $a$ be a solution to the equation $\sqrt{x^2 + 2} = \sqrt[3]{x^3 + 45}$. Evaluate the ratio of $\frac{2017}{a^2}$ to $a^2 - 15a + 2$.

A line $d$ is given in the plane. Let $B\in d$ and $A$ another point, not on $d$, and such that $AB$ is not perpendicular on $d$. Let $\omega$ be a variable circle touching $d$ at $B$ and letting $A$ outside, and $X$ and $Y$ the points on $\omega$ such that $AX$ and $AY$ are tangent to the circle. Prove that the line $XY$ passes through a fixed point. [i]Proposed by F. Bakharev [/i]

You are presented with a mystery function $f:\mathbb N^2\to\mathbb N$ which is known to satisfy \[f(x+1,y)>f(x,y)\quad\text{and}\quad f(x,y+1)>f(x,y)\] for all $(x,y)\in\mathbb N^2$. I will tell you the value of $f(x,y)$ for \$1. What's the minimum cost, in dollars, that it takes to compute the $19$th smallest element of $\{f(x,y)\mid(x,y)\in\mathbb N^2\}$? Here, $\mathbb N=\{1,2,3,\dots\}$ denotes the set of positive integers.

123456789=100. Here is the only way to insert 7 pluses and/or minus signs between the digits on the left side to make the equation correct: 1+2+3-4+5+6+78+9=100. Do this with only three plus or minus signs.

Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]

Calculate the following indefinite integrals. [1] $\int (\sin x+\cos x)^4 dx$ [2] $\int \frac{e^{2x}}{e^x+1}dx$ [3] $\int \sin ^ 4 xdx$ [4] $\int \sin 6x\cos 2xdx$ [5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$

A number is a palindrome if it does not change when the order of its digits is reversed. For example, $121$ and $23,432$ are palindromes. How many $4$-digit numbers are palindromes? $\text{(A) }9\qquad\text{(B) }10\qquad\text{(C) }45\qquad\text{(D) }90\qquad\text{(E) }100$

Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.

The numbers \( 1, 2, 3, \ldots, 60 \) are written in a row in that exact order. Igor and Ruslan take turns inserting the signs \( +, -, \times \) between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by $3$, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves? \\

Find the sum of the real roots of $2x^4 + 4x^3 + 6x^2 + 4x - 4$. [i]Individual #12[/i]

$x_1,...,x_n$ are reals and $x_1^2+...+x_n^2=1$ Prove, that exists such $y_1,...,y_n$ and $z_1,...,z_n$ such that $|y_1|+...+|y_n| \leq 1$; $max(|z_1|,...,|z_n|) \leq 1$ and $2x_i=y_i+z_i$ for every $i$

Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.

Compute $ax^{2018}+by^{2018}$, given that there exist real $a$, $b$, $x$, and $y$ which satisfy the following four equations: \begin{align*} ax^{2014}+by^{2014}&=6\\ ax^{2015}+by^{2015}&=7\\ ax^{2016}+by^{2016}&=3\\ ax^{2017}+by^{2017}&=50. \end{align*}

Benedek wrote the following $300 $ statements on a piece of paper. $2 | 1!$ $3 | 1! \,\,\, 3 | 2!$ $4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$ $5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$ $...$ $24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$ $25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$ How many true statements did Benedek write down? The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively. 1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$. 2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.