On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.

Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$.

Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

For $0 \leq p \leq 1/2,$ let $X_1, X_2, \ldots$ be independent random variables such that $$X_i=\begin{cases} 1 & \text{with probability } p, \\ -1 & \text{with probability } p, \\ 0 & \text{with probability } 1-2p, \end{cases} $$ for all $i \geq 1.$ Given a positive integer $n$ and integers $b,a_1, \ldots, a_n,$ let $P(b, a_1, \ldots, a_n)$ denote the probability that $a_1X_1+ \ldots + a_nX_n=b.$ For which values of $p$ is it the case that $$P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)$$ for all positive integers $n$ and all integers $b, a_1,\ldots, a_n?$

A positive integer is called [i]lonely [/i] if the sum of the inverses of its positive divisors (including $1$ and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by [i]Warren Bei[/i]

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$

A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable? [i]Proposed by Giacomo Rizzo[/i]

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

A sequence of integers $(a_0,...,a_k)$ is said to be [i]medaled[/i] if, for each $i = 0,...,k$, there are exactly $a_i$ elements of the sequence equal to $i$. For example, $(1,2,1,0)$ is a [i]medaled [/i] seqence. Indicates all [i]medaled [/i] sequences $(a_0,...,a_{2015})$.

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Don Miguel places a token in one of the $(n+1)^2$ vertices determined by an $n \times n$ board. A [i]move[/i] consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most $\sqrt2$ away, as long as it stays on the board. A [i]path[/i] is a sequence of moves such that the token was in each one of the $(n+1)^2$ vertices exactly once. What is the maximum number of diagonal moves (those of length $\sqrt2$) that a path can have in total?

A circle inscribed in a square, Has two chords as shown in a pair. It has radius $2$, And $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. [asy] size(100); defaultpen(linewidth(0.8)); draw(unitcircle); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); label("$A$",(-1,1),SE); label("$B$",(1,1),SE); label("$C$",(1,-1),SE); label("$D$",(-1,-1),SE); pair M=(1,0),N=(0,-1),T=(-1,0),U=(0,1),P=dir(135); draw(P--M^^(-1,-1)--(1,1)); label("$M$",M,SE); label("$N$",N,SE); label("$T$",T,SE); label("$U$",U,SE); label("$P$",P,dir(270)); dot(origin^^(-1,1)^^(-1,-1)^^(1,-1)^^(1,1)^^M^^N^^T^^U^^P); [/asy]

For a positive integer $n$, let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$. Find the value of $$A(T(2021))$$