Positive integer $k$ and $k$ colors are given. We will say that a set of $2k$ points on a plane is $colorful$, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of $k$ the least integer $n\geq 2$ such that: in every set of $nk$ points of a plane, no three of which are collinear, consisting of $n$ points of every color there exists a $colorful$ subset.

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

[b]Problem 1[/b] : Find the smallest positive integer $n$, such that $\sqrt[5]{5n}$, $\sqrt[6]{6n}$ , $\sqrt[7]{7n}$ are integers.

Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $BF = BE$ and such that $ED$ is the angle bisector of $\angle BEC$. Prove that $BD = EF$ if and only if $AF = EC$.

Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]

Prove that for all triangle $\triangle ABC$ holds the following inequality $$\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3$$

A circle is circumscribed around an isosceles triangle $ABC$ with base $BC$. The bisector of the angle $C$ and the bisector of the angles $A$ intersect the circle at the points $E$ and $D$, respectively, and the segment $DE$ intersects the sides $BC$ and $AB$ at the points $P$ and $Q$, respectively. Reconstruct $\vartriangle ABC$ given points $D, P, Q$, if it is known in which half-plane relative to the line $DQ$ lies the vertex $A$. (Maria Rozhkova)

A one-person game with two possible outcomes is played as follows: After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$

Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered $1$ through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50\%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each rounjd, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players $1$ and $2^n$ face each other in the last round? Express your answer in terms of $n$.

When filled with an cold water using a particular cold water tap, a tank will be full in 14 minutes. In 21 minutes, the full tank could be emptied by opening a hole on the base of the tank. If the cold water tap and the hot water tap are opened simultaneously (allowing hot and cold water fill the tank), and the hole on the base of the tank is opened, the tank will be full in 12.6 minutes. Determine the number of minutes needed to fill the tank with hot water until the tank is full, assuming at first the tank is empty and the hole is closed.

For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{one} \qquad \textbf{(C)}\ \text{two} \qquad \textbf{(D)}\ \text{more than two, but finitely many}\\ \textbf{(E)}\ \text{infinitely many}$

The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$? $ \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\ Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\ Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. [i]Proposed by Morteza Saghafian[/i]

Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$.

Show that for any positive integers $k$ and $1 \leq a \leq 9$, there exists $n$ such that satisfies the below statement. When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots $ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$.

Let $ABC$ be a triangle with incircle $\omega$ and incenter $I$. The circle $\omega$ is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. Point $P$ is the foot of the angle bisector from $A$ to $BC$, and point $Q$ is the foot of the altitude from $D$ to $EF$. Suppose $AI=7$, $IP=5$, and $DQ=4$. Compute the radius of $\omega$.

Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(5,2)$, is it possible to obtain the pair $(20,22)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(15,10)$, is it possible to obtain the pair $(27,23)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(5,8)$, is it possible to obtain the pair $(13,17)$ after a series of transformations?

In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\ 6, \ 7, \ 8,\ 9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?