There are two equal non-intersecting circles on a plane. Two lines were drawn. Each of the lines intersects the circles at four points so that the three segments formed on each of the lines are equal (segments with ends at adjacent points of intersection are considered). For one line these segments have length $a$, for the other they have length $b$ ($a < b$). Find the radius of the circles.

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$? [b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$. [b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points. [b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$. [b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins? [b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$ What is $a_{2011}$? [b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count. [b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$. [b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.

In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

A circle can be drawn around the quadrilateral $ABCD$. Let straight lines $AB$ and $CD$ intersect at point $M$, and straight lines $BC$ and $AD$ intersect at point $K$. (Points $B$ and $P$ lie on segments $AM$ and $AK$, respectively.) Let $P$ be the projection of point $M$ onto straight line $AK$, $L$ be the projection of point $K$ on the straight line $AM$. Prove that the straight line $LP$ divides the diagonal $BD$ in half.

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$? ${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$

The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$, where $[DEF]$ denotes the content of the triangle $DEF$. (David Hruska, Josef Tkadlec)

In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$. [i]Proposed by Zuming Feng[/i]

Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.

Let $ABC$ be a triangle, and let $\ell$ be the line passing through the circumcenter of $ABC$ and parallel to the bisector of the angle $\angle A$. Prove that the line $\ell$ passes through the orthocenter of $ABC$ if and only if $AB = AC$ or $\angle BAC = 120^o$

Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ . $(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$. $(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.

Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon? [asy] size(100); draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0)); draw((2,2)--(2,3)--(3,3)--(3,2)--cycle); [/asy]

Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. [i](Proposed by Danylo Khilko)[/i]

Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.

The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.

Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular. (B. Kukushkin)

Let $ABC$ be a triangle with $AB = AC$ ¸ $\angle BAC = 100^o$ and $AD, BE$ angle bisectors. Prove that $2AD <BE + EA$

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.