In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of  $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?

Several tiles congruent to the one shown in the picture below are to be fit inside a $11 \times 11$ square table, with each tile covering $6$ whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. [img]https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png[/img] Poland

Two triangles are said to be [i]twins [/i] if one of them is an image of the other one under a parallel projection. Prove that two triangles are twins if and only if either at least a side of one of them equals a side of another or both the triangles have equal segments that connect the corresponding vertices with some points on the opposite sides which divide these sides in the same ratio. (E. Barabanov)

Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$? [asy] draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--cycle); draw((0, 0)--(sqrt(3)/3, 1)); draw((0, 0)--(sqrt(3), 1)); label("A", (0, 1), N); label("B", (2, 1), N); label("C", (2, 0), S); label("D", (0, 0), S); label("E", (sqrt(3)/3, 1), N); label("F", (sqrt(3), 1), N); [/asy] ${ \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$

Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.

Let $H$ be an arbitrary point on the altitude $CP$ of the acute triangle $ABC$. The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$, respectively. [list] (a) Prove that $\angle NPC =\angle MPC$. (b) Let $O$ be the common point of $MN$ and $CP$. An arbitrary line through $O$ meets the sides of quadrilateral $CNHM$ in $D$ and $E$. Prove that $\angle EPC =\angle DPC$. [/list]

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

Positive numbers $a$, $b$, $c$ satisfy the inequalities \[a + b \geq ab, \quad b + c \geq bc,\quad\text{and}\quad c+ a \geq ca.\] Prove that $\displaystyle a + b + c \geq \frac34abc$.

Let $p$ be an odd prime and $x$ be an integer such that $p \mid x^3 - 1$ but $p \nmid x - 1$. Prove that \[ p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).\][i]John Berman[/i]

Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly. $a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$. $b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.

Given some red and blue points. Some of them are connected by the segments. Let us call "exclusive" the point, if its colour differs from the colour of more than half of the connected points. Every move one arbitrary "exclusive" point is repainted to the other colour. Prove that after the finite number of moves there will remain no "exclusive" points.

[b]p1.[/b] An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left(\frac{p}{4}\right)$ for $p > 61$ plates. a) For what number of plates $N$ does it become at least as cheap to use the second caterer as the first? b) Let $N$ be the number you found in a). For what number of plates $X$ is the second caterer's price exactly double the price for $N$ plates? c) Let $X$ be the number you found in b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer? [b]p2.[/b] Let $N$ be a positive integer. Prove the following: a) If $N$ is divisible by $4$, then $N$ can be expressed as the sum of two or more consecutive odd integers. b) If $N$ is a prime number, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. c) If $N$ is twice some odd integer, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. [b]p3.[/b] Let $S =\frac{1}{1^2} +\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$ a) Find, in terms of $S$, the value of $S =\frac{1}{2^2} +\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...$ b) Find, in terms of $S$, the value of$S =\frac{1}{1^2} +\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$ c) Find, in terms of $S$, the value of$S =\frac{1}{1^2} -\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...$ [b]p4.[/b] Let $\{P_1, P_2, P_3, ...\}$ be an infinite set of points on the $x$-axis having positive integer coordinates, and let $Q$ be an arbitrary point in the plane not on the $x$-axis. Prove that infinitely many of the distances $|P_iQ|$ are not integers. a) Draw a relevant picture. b) Provide a proof. [b]p5.[/b] Point $P$ is an arbitrary point inside triangle $ABC$. Points $X$, $Y$ , and $Z$ are constructed to make segments $PX$, $PY$ , and $PZ$ perpendicular to $AB$, $BC$, and $CA$, respectively. Let $x$, $y$, and $z$ denote the lengths of the segments $PX$, $PY$ , and $PZ$, respectively. a) If triangle $ABC$ is an equilateral triangle, prove that $x + y + z$ does not change regardless of the location of $P$ inside triangle ABC. b) If triangle $ABC$ is an isosceles triangle with $|BC| = |CA|$, prove that $x + y + z$ does not change when $P$ moves along a line parallel to $AB$. c) Now suppose that triangle $ABC$ is scalene (i.e., $|AB|$, $|BC|$, and $|CA|$ are all different). Prove that there exists a line for which $x+y+z$ does not change when $P$ moves along this line. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.

Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true? a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $

The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$. a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$. b) Find the locus of $P$ as we move the 3 arbitary lines.

Let $\triangle ABC$ be a triangle, and let $P_1,\cdots,P_n$ be points inside where no three given points are collinear. Prove that we can partition $\triangle ABC$ into $2n+1$ triangles such that their vertices are among $A,B,C,P_1,\cdots,P_n$, and at least $n+\sqrt{n}+1$ of them contain at least one of $A,B,C$.

Let $Q(n)$ denote the sum of the digits of $n$ in its decimal representation. Prove that for every positive integer $k$, there exists a multiple $n$ of $k$ such that $Q(n)=Q(n^2)$.