Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$. Find the minimum of $x^2+y^2+z^2+t^2$. [i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i]

Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.

Let $ABCDA'B'C'D'$ be a right parallelepiped, $E$ and $F$ the projections of $A$ on the lines $A'D$, $A'C$, respectively, and $P, Q$ the projections of $B'$ on the lines $A'C'$ and $A'C$ Prove that a) the planes $(AEF)$ and $(B'PQ)$ are parallel b) the triangles $AEF$ and $B'PQ$ are similar.

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2$ for $ k\equal{}1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.

In triangle $ABC$, the incircle touches sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Median of triangle $ABC$ from vertex $B$ meets segment $P Q$ at point $R$. Prove that angle $ARC$ is obtuse.

Let $ ABC$ be a triangle with $ \angle BAC \equal{} 90^\circ$. A circle is tangent to the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, such that the points on the circle diametrically opposite $ X$ and $ Y$ both lie on the side $ BC$. Given that $ AB \equal{} 6$, find the area of the portion of the circle that lies outside the triangle. [asy]import olympiad; import math; import graph; unitsize(20mm); defaultpen(fontsize(8pt)); pair A = (0,0); pair B = A + right; pair C = A + up; pair O = (1/3, 1/3); pair Xprime = (1/3,2/3); pair Yprime = (2/3,1/3); fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white); draw(A--B--C--cycle); draw(Circle(O, 1/3)); draw((0,1/3)--(2/3,1/3)); draw((1/3,0)--(1/3,2/3)); label("$A$",A, SW); label("$B$",B, down); label("$C$",C, left); label("$X$",(1/3,0), down); label("$Y$",(0,1/3), left);[/asy]

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that \[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p}, \]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.

Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.

Uri must paint some integers from $1$ to $2022$ (inclusive) in red, such that none of the differences between two red numbers is a prime number. Determine the maximum number of numbers Uri can paint red. [b]Note 1:[/b] The [i]difference [/i]between two distinct numbers is the subtraction of the larger minus the smaller. [b]Note 2:[/b] $1$ is not a prime number.

In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$. [i]Proposed by David Stoner[/i]

Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.

John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells. (A Shapovalov)