Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.

If $k$ raisins are distributed evenly to eleven children, four raisins would be left over. How many raisins would be left over if $3k$ raisins were distributed instead? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$

Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold: $\bullet ~~a_1\ge 2$. $\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$ $\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$ Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$

Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.

Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.

A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?

Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that $$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

In the sixth-year exams of a Center, they pass Physics at least$70\%$ of the students, Mathematics at least $75\%$; Philosophy at least, the $90\%$ and the Language at least, $85\%$. How many students, at least, pass these four subjects?

Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Suppose that there is a point $C$ which is equidistant from $A$ and $B$ at every moment. Prove that, at some moment, $A$ and $B$ will coincide.

[b](a)[/b] Prove that \[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [i]Author: Walther Janous, Austria[/i]

The squares of a $3\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, $2009$ is the label of the lower-rightmost square, and the label of each square divides the ne directly to the right of it and the one directly below it. How many such labelings are possible?

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

Cyclic quadrilateral $ABCD$ has side lengths $AB = 6$, $BC = 7$, $CD = 7$, $DA = 6$. What is the area of $ABCD$?

Prove that every section of the cube $Q = {[-1,1]}^n \subset R^n$ linear k-dimensional subspace $L\subseteq R^n$ has a diameter of at least $2\sqrt k$.

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ $\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85$

Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$

Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$

Let $f: N \to N$ satisfy $n=\sum_{d|n} f(d), \forall n \in N$. Then sum of all possible values of $f(100)$ is?

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Let $ABC$ be an acute-angled triangle such that $\angle{ACB} = 45^{\circ}$. Let $G$ be the point of intersection of the three medians and let $O$ be the circumcentre. Suppose $OG=1$ and $OG \parallel BC$. Determine the length of the segment $BC$.

$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

In the triangle $ABC$ it is known that$\angle A = 75^o, \angle C = 45^o$. On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$. Let $M$ be the midpoint of the segment $AT$. Find the measure of the $\angle BMC$. (Anton Trygub)

In a tennis tournament there are participants from $n$ different countries. Each team consists of a coach and a player whom should settle in a hotel. The rooms considered for the settlement of coaches are different from players' ones. Each player wants to be in a room whose roommates are [b][u]all[/u][/b] from countries which have a defense agreement with the player's country. Conversely, each coach wants to be in a room whose roommates are [b][u]all[/u][/b] from countries which don't have a defense agreement with the coach's country. Find the minimum number of the rooms such that we can [u][b]always[/b][/u] grant everyone's desire. [i]proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi[/i]

Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved. Prove that the triangle $XYZ$ is completely covered by the expanded disks.