The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Let $p$ be a prime number. Let $\mathbb F_p$ denote the integers modulo $p$, and let $\mathbb F_p[x]$ be the set of polynomials with coefficients in $\mathbb F_p$. Define $\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$ by \[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \] Prove that for nonzero polynomials $F,G \in \mathbb F_p[x]$, \[ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). \] Here, a polynomial $Q$ divides $P$ if there exists $R \in \mathbb F_p[x]$ such that $P(x) - Q(x) R(x)$ is the polynomial with all coefficients $0$ (with all addition and multiplication in the coefficients taken modulo $p$), and the gcd of two polynomials is the highest degree polynomial with leading coefficient $1$ which divides both of them. A non-zero polynomial is a polynomial with not all coefficients $0$. As an example of multiplication, $(x+1)(x+2)(x+3) = x^3+x^2+x+1$ in $\mathbb F_5[x]$. [i]Proposed by Mark Sellke[/i]

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.

[b]2.[/b] Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$, with $\int_{0}^{1}f_{1}(x) dx= 0$ $ (i=1,\dots ,n)$. Show that, for every $\alpha \in (0,1)$, there existis a subset $E$ of $[0,1]$ with measure $\alpha$, such that $\int_{E}f_{i}(x)dx=0$. [b](R. 17)[/b]

There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear. A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions: $f(x+y) < f(x) + f(y)$ $f(f(x)) = \lfloor {x} \rfloor + 2$

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\in S$. For each integer $n\ge 2$, find the largest integer $k$ (in terms of $n$) such that there exists a set of $n$ distinct points that is [i]not[/i] overdetermined, but has $k$ overdetermined subsets. [i]Proposed by Carl Schildkraut[/i]