The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$, respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.

Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.

A positive integer $n > 1$ is juicy if its divisors $d_1 < d_2 < \dots < d_k$ satisfy $d_i - d_{i-1} \mid n$ for all $2 \leq i \leq k$. Find all squarefree juicy integers.

Is it possible to assign every integral point $(x,y)$ of the plane a positive integer $a_{x,y}$ such that for every two integers $i$ and $j$ the following equality holds $$a_{i,j}=\gcd(a_{i-1,j},a_{i+1,j})+\gcd(a_{i,j-1},a_{i,j+1})$$ [i]M. Shutro[/i]

A domino is a rectangle whose length is twice its width. Any square can be divided into seven dominoes, for example as shown in the figure below. [img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img] a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$. b) Show that you cannot divide a square into three or four dominoes.

How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)

Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.

Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that $a)$ $\alpha$ is real and negative; $b)$ the remaining third quadratic equation has non-real roots.

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

Let it \(C,D > 0\). We call a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) [i]pretty[/i] if \(f\) is a \(C^2\)-class, \(|x^3f(x)| \leq C\) and \(|xf''(x)| \leq D\). [list='i'] [*] Show that if \(f\) is pretty, then, given \(\epsilon \geq 0\), there is a \(x_0 \geq 0\) such that for every \(x\) with \(|x| \geq x_0\), we have \(|x^2f'(x)| < \sqrt{2CD}+\epsilon\). [*] Show that if \(0 < E < \sqrt{2CD}\) then there is a pretty function \(f\) such that for every \(x_0 \geq 0\) there is a \(x > x_0\) such that \(|x^2f'(x)| > E\). [/list]

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then $$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$

Two triangles $ACE, BDF$ are given which intersect at six points: $G, H, I, J, K, L$ as in the picture. It is known that in each of the quadrilaterals \[ABIK ,BCJL ,CDKG ,DELH ,EFGI\] it is possible to inscribe a circle. Is it possible for the quadrilateral $FAHJ$ is also circumscribed around a circle?

Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A, B$ and $C$ intersect $\Omega$ at $A_1, B_1$ and $C_1$, respectively, and the internal bisectors of angles $A_1, B_1$ and $C_1$ of the triangles $A_1 A_2 A_ 3$ intersect $\Omega$ at $A_2, B_2$ and $C_2$, respectively. If the smallest angle of the triangle $ABC$ is $40^{\circ}$, what is the magnitude of the smallest angle of the triangle $A_2 B_2 C_2$ in degrees?

Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.

Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

If $3a=1+\sqrt 2$, what is the largest integer not exceeding $9a^4-6a^3+8a^2-6a+9$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the preceding} $

Given distinct real numbers $a_1,a_2,...,a_n$, find the minimum value of the function $$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$

Given $101$ segments in a line, prove that there exists $11$ segments meeting in $1$ point or $11$ segments such that every two of them are disjoint.

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the attached figure, point $C$ is the center of the circle, $AB$ is tangent to the circle, $P-C-P'$ and $AC\perp PP'$. If $AT = 2$ cm. and $AB = 4$ cm, calculate $BQ$ [img]https://cdn.artofproblemsolving.com/attachments/e/e/d47429b82fb87299c40f5224489313909cfd0f.png[/img] Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.