$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $

A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment. Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$ Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide. [i]Tilek Askerbekov[/i]

There are three straight roads. Three pedestrians are moving along those roads, and they are NOT on one line in the initial moment. Prove that they will be one line not more than twice

1. Prove the following inequality for positive reals $a_1,a_2...,a_n$ and $b_1,b_2...,b_n$: $(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})$

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Some students of a group were friends of some others. One day all students of the group take part in a picnic. During the picnic some friends had a quarrel with each other, but some other students became friends. After the picnic, the number of friends for each student changed by $1$. Prove that the number of students in the group was even.

Determine all natural numbers n for which the number $A = n^4 + 4n^3 +5n^2 + 6n$ is a perfect square of a natural number.

A set is called $six\ square$ if it has six pair-wise coprime numbers and for any partition of it into two set with three elements, the sum of the numbers in one of them is perfect square. Prove that there exist infinitely many $six\ square$.

There are $3\leq n\leq 25$ passengers in a bus some of which are friends. Every passenger has exactly $k$ friends among the passengers, no two friends have a common friend and every two people, who are not friends have a common friend. Find $n$.

[url=https://artofproblemsolving.com/community/c678145][b]Czech-Polish-Slovak Match 2018[/b][/url] [b]Austria, 24 - 27 June 2018[/b] [url=http://artofproblemsolving.com/community/c6h1667029p10595005][b]Problem 1.[/b][/url] Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i] [url=http://artofproblemsolving.com/community/c6h1667030p10595011][b]Problem 2.[/b][/url] Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$. [i]Proposed by Patrik Bak, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667031p10595016][b]Problem 3.[/b][/url] There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns. [i]Proposed by Peter Novotný, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667033p10595021][b]Problem 4.[/b][/url] Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i] [url=http://artofproblemsolving.com/community/c6h1667034p10595023][b]Problem 5.[/b][/url] In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667036p10595032][b]Problem 6.[/b][/url] We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ [b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic. [b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic. [i]Proposed by Walther Janous, Austria[/i]

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

If $ \angle\text{CBD} $ is a right angle, then this protractor indicates that the measure of $ \angle\text{ABC} $ is approximately [asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E);[/asy] $ \text{(A)}\ 20^\circ\qquad\text{(B)}\ 40^\circ\qquad\text{(C)}\ 50^\circ\qquad\text{(D)}\ 70^\circ\qquad\text{(E)}\ 120^\circ $

There are integers $a$ and $b$, such that $a>b>1$ and $b$ is the largest divisor of $a$ different from $a$. Prove that the number $a+b$ is not a power of $2$ with integer exponent.

For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.

$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

On an infinite “squared” sheet six squares are shaded as in the diagram. On some squares there are pieces. It is possible to transform the positions of the pieces according to the following rule: if the neighbour squares to the right and above a given piece are free, it is possible to remove this piece and put pieces on these free squares. The goal is to have all the shaded squares free of pieces. Is it possible to reach this goal if (a) In the initial position there are $6$ pieces and they are placed on the $6$ shaded squares? (b) In the initial position there is only one piece, located in the bottom left shaded square? [img]https://cdn.artofproblemsolving.com/attachments/2/d/0d5cbc159125e2a84edd6ac6aca5206bf8d83b.png[/img] (M Kontsevich, Moscow)

Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that \[ f(ab) = f(a)b + af(b) \] for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$? [i] (Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.)[/i]

Prove that no matter what digits are placed in the four empty boxes, the eight-digit number \[ \textbf{9999}\Box\Box\Box\Box \] is not a perfect square. (A $\textit{perfect square}$ is a whole number times itself. For example, $25$ is a perfect square because $25 = 5 \times 5$.)

Find all real numbers $a$ such that $4 < a < 5$ and $a(a-3\{a\})$ is an integer. ({x} represents the fractional part of x)

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

Let $ABC$ be an acute triangle and $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. Call $Y$ and $Z$ the feet of the perpendicular lines from $B$ and $C$ to $FD$ and $DE$, respectively. Let $F_1$ be the symmetric of $F$ with respect to $E$ and $E_1$ be the symmetric of $E$ with respect to $F$. If $3EF=FD+DE$, prove that $\angle BZF_1=\angle CYE_1$.

In right triangle $ABC$, let $D$ be the midpoint of hypotenuse $[AB]$, circumradius be $\dfrac 52$ and $|BC|=3$. What is the distance between circumcenter of $\triangle ACD$ and incenter of $\triangle BCD$? $ \textbf{(A)}\ \dfrac {29}{2} \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ \dfrac 52 \qquad\textbf{(D)}\ \dfrac{5\sqrt{34}}{12} \qquad\textbf{(E)}\ 2\sqrt 2 $

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrović