A new chess piece named $ mammoth $ moves like a bishop l (that is, in a diagonal), but only in 3 of the 4 possible directions. Different $ mammoths $ in the board may have different missing addresses. Find the maximum number of $ mammoths $ that can be placed on an $ 8 \times 8 $ chessboard so that no $ mammoth $ can be attacked by another.

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

Given a triangle $ABC$, on the side $BC$ which marked the point $E$ such that $BE \ge CE$. Construct on the sides $AB$ and $AC$ the points $D$ and $F$, respectively, such that $\angle DEF = 90 {} ^ \circ$ and the segment $BF$ is bisected by the segment $DE $. (Black Maxim)

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$

Which compound contains the highest percentage of nitrogen by mass? $ \textbf{(A)} \text{NH}_2\text{OH} (M=33.0) \qquad\textbf{(B)}\text{NH}_4\text{NO}_2 (M=64.1)\qquad$ $\textbf{(C)}\text{N}_2\text{O}_3 (M=76.0)\qquad\textbf{(D)}\text{NH}_4\text{NH}_2\text{CO}_2 (M=78.1)\qquad $

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.

Let $A=[0,0]$ and $B=[n,n]$. In how many ways can we go from $A$ to $B$, if we always want to go from lattice point to its neighbour (i.e. point with one coordinate the same and one smaller or bigger by one), we never want to visit the same point twice and we want our path to have length $2n+2$? (For example, path $[0,0],[0,1],[-1,1],[-1,2],[0,2],[1,2],[2,2],[2,3],[3,3]$ is one of the paths for $n=3$)

Given right hexagon $H$ with side length $1$. On the sides of $H$ points $A_1$,$A_2$,$\ldots$,$A_k$ such that at least one of them is the midpoint of some side and for every $1 \leq i \leq k$ lines $A_{i-1}A_i$ and $A_iA_{i+1}$ form equal angles with the side, that contains the point $A_i$ (let $A_0=A_k$ and $A_{k+1}=A_1$. It is known that the length of broken line $A_1A_2\ldots A_kA_1$ is a positive integer Prove that $n$ is divisible by $3$ [i]M. Zorka[/i]

Prove that if a graph $\mathcal{G}$ on $n\ge 3$ vertices has a unique $3$-coloring, then $\mathcal{G}$ has at least $2n-3$ edges. (A graph is $3$-colorable when there exists a coloring of its vertices with $3$ colors such that no two vertices of the same color are connected by an edge. The graph can be $3$-colored uniquely if there do not exist vertices $u$ and $v$ of the graph that are painted different colors in one $3$-coloring, yet are colored the same in another.)

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]