Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors.

Let $C$ be a circle, and let $A, B, P$ be three points on $C$. Let $L_A$ and $L_B$ be the tangent lines to $C$ passing through $A$ and $B$, respectively. Let $a$ and $b$ be the distances from $P$ to $L_A$ and $L_B$, respectively, and let $c$ be the distance from $P$ to the chord of $C$ determined by $A$ and $B$. Prove that $c^2 = a \cdot b$.

In the triangle $ABC$ with sides $BC> AC> AB$ the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three. (Rozhkova Maria)

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Prove the inequality $ \sqrt {2 \plus{} \sqrt [3]{3 \plus{} ... \plus{} \sqrt [{2008}]{2008}}} < 2$

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

Find the maximum possible value of $a + b + c ,$ if $a,b,c$ are positive real numbers such that $a^2 + b^2 + c^2 = a^3 + b^3 + c^3 .$

$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60$ Prove, that $AB>AC$

The incircle of $ \triangle A_{0}B_{0}C_{0}$, meets legs $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$, respectively on points $A$, $B$, $C$, and the incircle of $ \triangle ABC$, with center $I$, meets legs $BC$, $CA$, $AB$, on points $A_{1}$, $B_{1}$, $C_{1}$, respectively. We write with $ \sigma (ABC)$, and $ \sigma (A_{1}B_{1}C_{1})$ the areas of $ \triangle ABC$, and $ \triangle A_{1}B_{1}C_{1}$ respectively. Prove that if $ \sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})$, then lines $AA_{0}$, $BB_{0}$, $CC_{0}$ are concurrent.

Determine all functions $f:\mathbb R\to\mathbb R$ such that equality $$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$ holds for all real numbers $x$, $y$. Proposed by Athanasios Kontogeorgis

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$

Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

Solve the following in equation in $\mathbb{R}^3$: \[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]

Let $P$ be the midpoint of the arc $AC$ of a circle, and $B$ be a point on the arc $AP$. Let $M$ and $N$ be the projections of $P$ onto the segments $AC$ and $BC$ respectively. Prove that if $D$ is the intersection of the bisector of $\angle ABC$ and the segment $AC$, then every diagonal of the quadrilateral $BDMN$ bisects the area of the triangle $ABC$.

Let $a$, $b$, $n$ be positive integers such that $a + b \leq n^2$. Alice and Bob play a game on an (initially uncoloured) $n\times n$ grid as follows: - First, Alice paints $a$ cells green. - Then, Bob paints $b$ other (i.e.uncoloured) cells blue. Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of $a$, $b$ and $n$, who has a winning strategy.

Let $X$ be a set with $n$ elements, and let $A_{1}$, $A_{2}$, ..., $A_{m}$ be subsets of $X$ such that: 1) $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$; 2) $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$. Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$, the set $A$ does not contain $A_{i}$. [i]Alternative formulation.[/i] Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$, such that $|A_{i}\cap A_{j}|\leq 1$, for every $i\neq j$. Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$, such that none of $A_{i}$'s is a subset of $A$.