Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ : \[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]

Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$. Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that \[\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.\] [i]Proposed by Orif Ibrogimov, Uzbekistan.[/i]

The persons $P_1, P_2, . . . , P_{n-1}, P_n$ sit around a table, in this order, and each one of them has a number of coins. In the start, $P_1$ has one coin more than $P_2, P_2$ has one coin more than $P_3$, etc., up to $P_{n-1}$ who has one coin more than $P_n$. Now $P_1$ gives one coin to $P_2$, who in turn gives two coins to $P_3 $ etc., up to $ Pn$ who gives n coins to $ P_1$. Now the process continues in the same way: $P_1$ gives $n+ 1$ coins to $P_2$, $P_2$ gives $n+2$ coins to $P_3$; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.

Let $\omega_1$ and $\omega_2$ be circles with radii $3$ and $12$ and externally tangent at point $P$. Let a common external tangent intersect $\omega_1$, $\omega_2$ at $S$, $T$ respectively and the common internal tangent at point $Q$. Define $X$ to be the point on $\overrightarrow{QP}$ such that $QX=10$. If $XS$, $XT$ intersect $\omega_1$, $\omega_2$ a second time at $A$ and $B$, determine $\tan\angle APB$. . [i]2015 CCA Math Bonanza Individual Round #15[/i]

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $ \lim _ { x \rightarrow \infty } f ^ { \prime } ( x ) = 0 $. Prove that $ \lim _ { x \rightarrow \infty } f ( x ) / x = 0 $.

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.