Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

Prove that any quadratic expression $$Q(x) = Ax^2 + Bx + C$$ (a) can be put into the form $$Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m$$ where $k, \ell, m$ depend on the coefficients $A,B,C$ and (b) $Q(x)$ takes on integral values for every integer $x$ if and only if $k, \ell, m$ are integers.

Pablo and Nacho write together a succession of positive integers of $2006$ terms, according to the following rules: Pablo begins, who in his first turn writes $1$, and from then on, each one in his turn writes an integer positive that is greater than or equal to the last number that the opponent wrote and less than or equal to triple the last number that the opponent wrote. When the two of them have written the $2006$ numbers, the sum $S$ of the first $ 2005$ numbers written (all except the last one) and the sum $T$ of the $2006$ numbers written. If $S$ and $T $ are co-cousins, Nacho wins. Otherwise, Pablo wins. Determine which of the two players has a winning strategy, describe the strategy and demonstrate that it is a winning one.

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent. Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $

A merchant buys a number of oranges at $ 3$ for $ 10$ cents and an equal number at $ 5$ for $ 20$ cents. To "break even" he must sell all at: $ \textbf{(A)}\ \text{8 for 30 cents} \qquad \textbf{(B)}\ \text{3 for 11 cents} \qquad \textbf{(C)}\ \text{5 for 18 cents} \\ \textbf{(D)}\ \text{11 for 40 cents} \qquad \textbf{(E)}\ \text{13 for 50 cents}$

A positive integer $N$ is \emph{piquant} if there exists a positive integer $m$ such that if $n_i$ denotes the number of digits in $m^i$ (in base $10$), then $n_1+n_2+\cdots + n_{10}=N$. Let $p_M$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim\limits_{M\to \infty} p_M$. [i]Proposed by James Lin.[/i]

Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

Let $f(x)=x^2-\pi x$, $\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)$ $\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$ $\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$ $\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$ $\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)$

Three points $P_1, P_2,$ and $P_3$ and three lines $\ell_1, \ell_2,$ and $\ell_3$ lie in the plane such that none of the three points lie on any of the three lines. For (not necessarily distinct) integers $i$ and $j$ between 1 and 3 inclusive, we call a line $\ell$ $(i, j)$-[i]good[/i] if the reflection of $P_i$ across $\ell$ lies on $\ell_j$, and call it [i]excellent[/i] if there are two distinct pairs $(i_1, j_1)$ and $(i_2, j_2)$ for which it is good. Suppose that exactly $N$ excellent lines exist. Compute the largest possible value of $N$. [i]Proposed by Yannick Yao[/i]

Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations $$\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}$$

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

The positive integers $a$, $p$, $q$ and $r$ are greater than $1$ and are such that $p$ divides $aqr+1$, $q$ divides $apr+1$ and $r$ divides $apq+1$. Prove that: a) There are infinitely many such quadruples $(a,p,q,r)$. b) For each such quadruple we have $a\geq \frac{pqr-1}{pq+qr+rp}$.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.

The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? $\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$

Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$. What is the probability of drawing a number from the bag that is divisible by $11$?

Determine the least integer $k$ for which the following story could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$.

$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation $\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$ Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: ${{ \textbf{(A)}\ \text{infinite} \qquad\textbf{(B)}\ \text{finite, but greater than 2} \qquad\textbf{(C)}\ \text{Two} \qquad\textbf{(D)}\ \text{One} }\qquad\textbf{(E)}\ \text{Zero} } $

Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.