Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.

Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows: K_0=1 K_1 (x_1 )=〖x_1〗^2 K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n ) Prove that: K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.

Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$. Proposed by: L Shatunov

Let $ a,b,c$ be real positive numbers such that absolute difference between any two of them is less than $ 2$. Prove that: $ a \plus{} b \plus{} c < \sqrt {ab \plus{} 1} \plus{} \sqrt {ac \plus{} 1} \plus{} \sqrt {bc \plus{} 1}$

Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.

Find three consecutive odd numbers $a,b,c$ such that $a^2+b^2+c^2$ is a four digit number with four equal digits.

Find the smallest $n$ such that $2^{2000}$ divides $n!$.

In the plane with orthogonal Cartesian coordinates $x$ and $y$, prove that the line whose equation is $ux+vy = 1$ will be tangent to the cirve $x^m+y^m=1$ (where $m>1$) if and only if $u^n + v^n = 1$ and $m^{-1} + n^{-1} = 1$.

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? $\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$

On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: [list] [*] $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers; [*] $(2)$ there are only zeros on the blackboard. [/list] Player $B$ must then give as many cookies to player $A$ as there are numbers on the blackboard. Player $A$ wants to get as many cookies as possible, whereas player $B$ wants to give as few as possible. Determine the number of cookies that $A$ receives if both players play optimally.

Let be given a positive integer $n > 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that $$P(x)P(x^2) ... P(x^n) = P\left( x^{\frac{n(n+1)}{2}}\right)$$ for all $x \in R$

You are given that \[17! = 355687ab8096000\] for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two nondecreasing functions. [list=a] [*]Show that for any $a\in\mathbb{R},$ $b\in[f(a-0),f(a+0)]$ and $x\in\mathbb{R},$ the following inequality holds \[\int_a^xf(t) \ dt\geq b(x-a).\] [*]Given that $[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset$ for any $a\in\mathbb{R},$ prove that for any real numbers $a<b$\[\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.\] [/list] [i]Note: $h(a-0)$ and $h(a+0)$ denote the limits to the left and to the right respectively of a function $h$ at point $a\in\mathbb{R}.$[/i]

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.

A group consists of $n$ tourists. Among every three of them there are at least two that are not familiar. For any partition of the group into two busses, there are at least two familiar tourists in one bus. Prove that there is a tourist who is familiar with at most two fifth of all the tourists.

Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$: $$\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.$$ Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$. [i]Remark:[/i] A sequence $x_1,x_2,x_3,\dotsc $ is [i]eventually periodic[/i] if there are positive integers $c$ and $t$ such that $x_n=x_{n+t}$ for all $n\geqslant c$. [i]Proposed by Petar Nizié-Nikolac[/i]

Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column. What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?