A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.

Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences \begin{eqnarray*} P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\ Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014) \end{eqnarray*} are permutations of each other. (a) Prove that there exist distinct block-similar polynomials of degree $n + 1$. (b) Prove that there do not exist distinct block-similar polynomials of degree $n$. [i]Proposed by David Arthur, Canada[/i]

(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$. (b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$? (A Shapovalov) PS. part (a) for Juniors, both part for Seniors

Compute the number of positive integers $n$ less than $100$ for which $1+2+\dots+n$ is not divisible by $n.$

Simplify: $ \sum\limits_{k \equal{} 10}^{2006} \binom{k}{10}$ (Your answer should contain no summations but may still contain binomial coefficients/combinations.)

Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.

An angle with vertex $ A$ and measure $ \alpha$ and a point $ P_0$ on one of its rays are given so that $ AP_0\equal{}2$. Point $ P_1$ is chose on the other ray. The sequence of points $ P_1,P_2,P_3,...$ is defined so that $ P_n$ lies on the segment $ AP_{n\minus{}2}$ and the triangle $ P_n P_{n\minus{}1} P_{n\minus{}2}$ is isosceles with $ P_n P_{n\minus{}1}\equal{}P_n P_{n\minus{}2}$ for all $ n \ge 2$. $ (a)$ Prove that for each value of $ \alpha$ there is a unique point $ P_1$ for which the sequence $ P_1,P_2,...,P_n,...$ does not terminate. $ (b)$ Suppose that the sequence $ P_1,P_2,...$ does not terminate and that the length of the polygonal line $ P_0 P_1 P_2 ... P_k$ tends to $ 5$ when $ k \rightarrow \infty$. Compute the length of $ P_0 P_1$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]

Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects * the line $AI$ in points $A$ and $P$ * the line $BI$ in points $B$ and $Q$ * the line $AC$ in points $A$ and $R$ * the line $BC$ in points $B$ and $S$ with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively. Prove that the lines $PS$, $QR$, and $CI$ meet in a single point. (Stephan Wagner)

A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.

All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]

To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not ipped or rotated) on his plate and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed?

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so.

Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.

For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\ r^2 & (r\leq x\leq l \plus{} r)\quad \\ (x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$ Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]

Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is $\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$