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.

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]

If $ x$, $ y$, $ z$ are nonnegative real numbers with the sum $ 1$, find the maximum value of $ S \equal{} x^2(y \plus{} z) \plus{} y^2(z \plus{} x) \plus{} z^2(x \plus{} y)$ and $ C \equal{} x^2y \plus{} y^2z \plus{} z^2x$.

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$. (Peter Novotný)

Let $p,\ q$ be positive integers. Consider integrs $a,\ b,\ c$ satisfying conditions: \[-q\leq b\leq 0\leq a\leq p,\ \ b\leq c\leq a\] Call such $a,\ b,\ c$ in arranging in the form of $[a,\ b\ ;\ c]$ as $(p,\ q)$ pattern. For each $(p,\ q)$ pattern $[a,\ b\ ;\ c]$, let $w([a,\ b\ ;\ c])=p-q-(a+b)$. (1) Find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=-q$, then find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=p$. From now on, we consider the case of $p=q$. (2) Let $s$ be integer. Find the number of $(p,\ p)$ pattern such that $w([a,\ b\ ;\ c])=-p+s$. (3) Find the total number of $(p,\ p)$ pattern. [i]2011 Tokyo University entrance exam/Science, Problem 5[/i]

On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different. Find the smallest possible amount of the sum of all written numbers.

Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.

We will call a pair of positive integers $(n, k)$ with $k > 1$ a $lovely$ $couple$ if there exists a table $nxn$ consisting of ones and zeros with following properties: • In every row there are exactly $k$ ones. • For each two rows there is exactly one column such that on both intersections of that column with the mentioned rows, number one is written. Solve the following subproblems: a) Let $d \neq 1$ be a divisor of $n$. Determine all remainders that $d$ can give when divided by $6$. b) Prove that there exist infinitely many lovely couples. Proposed by Miroslav Marinov, Daniel Atanasov

In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.

Given $5$ points of a sphere radius $r$, show that two of the points are a distance $\le r \sqrt2$ apart.

How many non-empty subsets of $\{1, 2, 3, 4, 5, 6,7,8\}$ have exactly $k$ elements and do not contain the element $k$ for some $k = 1, 2,...,8$.

At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle? $ \textbf{(A)}\ 3.75 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8.75 $

The captain distributed $4000$ gold coins among $40$ pirates. A group of $5$ pirates is called poor if those $5$ pirates received, together, $500$ coins or less. The captain made the distribution so that there were the minimum possible number of poor groups of $5$ pirates. Determine how many poor $5$ pirate groups there are. Clarification: Two groups of $5$ pirates are considered different if there is at least one pirate in one of them who is not in the other.

Let be $x_1=\frac{1}{\sqrt[n+1]{n!}}$ and $x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}$ for all $n\in \mathbb{N}^*$ and $f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}$ where $$f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}$$Prove that the sequence $(a_n)_{n\geq1}$ when $a_n=\int \limits_{x_1}^{x_2}f(x)dx$ is convergent and compute $$\lim \limits_{n \to \infty}a_n$$

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle in a given direction (that is, the numbers $a$, $b$, $c$, $d$ are replaced by $a-b$, $b-c$, $c-d$, $d-a$). Is it possible after $1996$ such steps to have numbers $a$, $b$, $c$ and $d$ such that the numbers $|bc-ad|$, $|ac-bd|$ and $|ab-cd|$ are primes?

Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.

A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle $ABC$ lie on this parabola so that the bisector of angle $\angle BAC$ is parallel to the axis $Ox$ . Prove that the midpoint of the median drawn from vertex $A$ lies on the axis of the parabola.

Find all integer solutions of the equation \[1 + x + x^2 + x^3 + x^4 = y^4.\]

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]

A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$. Prove that $XN=NY$.

On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.

Give a point set on a plane $P=\{P_1,P_2,\cdots,P_{1994}\}$. Any three points in $P$ are not colinear. Divide points in $P$ into $83$ groups, in each group there are at least three points, and any point exactly belongs to one group. Connect any two points in the same group with a line segment, but we do not connect points not in the same group. Now we get a figure $G$. Note the number of triangles in $G$ : $m(G)$. [b](a)[/b] Find the minumum value of $m(G)$ : $m_0$. [b](b)[/b] $G*$ is one of the figures that $m(G)=m_0$, color the line segments in $G*$ with four colors. Prove that there exists a proper way, satisfying that no triangle is made up of three sides in the same color.

The positive integer $ n$ has the following property: if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains. Find $n$.

Show that there exist no integer solutions $(x, y, z)$ to the equation $$x^3+2y^3+4z^3=9$$