The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

Let $ABCD$ be a convex and cyclic quadrilateral. Let $AD\cap BC=\{E\}$, and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$. Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.

Let $\delta_0=\triangle A_0B_0C_0$ be a triangle. On each of the sides $B_0C_0$, $C_0A_0$, $A_0B_0$, there are constructed squares in the halfplane, not containing the respective vertex $A_0,B_0,C_0$ and $A_1,B_1,C_1$ are the centers of the constructed squares. If we use the triangle $\delta_1=\triangle A_1B_1C_1$ in the same way we may construct the triangle $\delta_2=\triangle A_2B_2C_2$; from $\delta_2=\triangle A_2B_2C_2$ we may construct $\delta_3=\triangle A_3B_3C_3$ and etc. Prove that: (a) segments $A_0A_1,B_0B_1,C_0C_1$ are respectively equal and perpendicular to $B_1C_1,C_1A_1,A_1B_1$; (b) vertices $A_1,B_1,C_1$ of the triangle $\delta_1$ lies respectively over the segments $A_0A_3,B_0B_3,C_0C_3$ (defined by the vertices of $\delta_0$ and $\delta_1$) and divide them in ratio $2:1$. [i]K. Dochev[/i]

Find all positive integers $d$ that can be written in the form $$ d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|), $$ where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$, $y^2 \neq z$, and $z^2 \neq x$.

Find all the real numbers $x$ that verify the equation: $$x-3\{x\}-\{3\{x\}\}=0$$ $\{a\}$ represents the fractional part of $a$

The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of 100, the population was one more than a perfect square. Now, with an additional increase of 100, the population is again a perfect square. The original population is a multiple of: $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 17$

Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

For a natural number $N$, consider all distinct perfect squares that can be obtained from $N$ by deleting one digit from its decimal representation. Prove that the number of such squares is bounded by some value that doesn't depend on $N$.

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

In rectangle $ABCD$, $AB = 7$ and $AC = 25$. What is its area?

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that atleast $15$ balls of a single colour will be drawn?

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.

The numbers $a_1, a_2, ..., a_n$ are equal to $1$ or $-1$. Prove that $$2 \sin \left(a_1+\frac{a_1a_2}{2}+\frac{a_1a_2a_3}{4}+...+\frac{a_1a_2...a_n}{2^{n-1}}\right)\frac{\pi}{4}=a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+...+a_n\sqrt2}}}$$ In particular, for $a_1 = a_2 = ... = a_n = 1$ we have $$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?

In any set of $181$ square integers, prove that one can always find a subset of $19$ numbers, sum of whose elements is divisible by $19$.

Let triangle $ ABC$ acutangle, with $ m \angle ACB\leq\ m \angle ABC$. $ M$ the midpoint of side $ BC$ and $ P$ a point over the side $ MC$. Let $ C_{1}$ the circunference with center $ C$. Let $ C_{2}$ the circunference with center $ B$. $ P$ is a point of $ C_{1}$ and $ C_{2}$. Let $ X$ a point on the opposite semiplane than $ B$ respecting with the straight line $ AP$; Let $ Y$ the intersection of side $ XB$ with $ C_{2}$ and $ Z$ the intersection of side $ XC$ with $ C_{1}$. Let $ m\angle PAX \equal{} \alpha$ and $ m\angle ABC \equal{} \beta$. Find the geometric place of $ X$ if it satisfies the following conditions: $ (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP}$ $ (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}$

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.

A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?

Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup … \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]