Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $ABCD$ be a convex quadrilateral with $m(\widehat{ADC}) = 90^{\circ}$. The line through $D$ which is parallel to $BC$ meets $AB$ at $E$. If $m(\widehat{DAC}) = m(\widehat{DAE})$, $|AB|=3$ and $|AC|=4$, then $|AE| = ?$ $\textbf{(A)}\ \frac56 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \frac34$

On the sides of triangle $ABC$ take the points $M_1, N_1, P_1$ such that each line $MM_1, NN_1, PP_1$ divides the perimeter of $ABC$ in two equal parts ($M, N, P$ are respectively the midpoints of the sides $BC, CA, AB$). [b]I.[/b] Prove that the lines $MM_1, NN_1, PP_1$ are concurrent at a point $K$. [b]II.[/b] Prove that among the ratios $\frac{KA}{BC}, \frac{KB}{CA}, \frac{KC}{AB}$ there exist at least a ratio which is not less than $\frac{1}{\sqrt{3}}$.

The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that: [b](a)[/b] $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$ [b](b)[/b] $\angle ACB \geq 60^o$.

The incircle $\omega$ of triangle $ABC$, right angled at $C$, touches the circumcircle of its medial triangle at point $F$. Let $OE$ be the tangent to $\omega$ from the midpoint $O$ of the hypotenuse $AB$, distinct from $AB$. Prove that $CE = CF$.

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that: (a) no two rooks attack one another, and (b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks. [img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img] The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has. [b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees. [b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$. [b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$. [b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is, $$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$ Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$. [b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.) [b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$. [b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$. [b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\vartriangle XY Z$ be a triangle such that $\angle X = 70^o$. There exists a point $F$ inside triangle $\vartriangle XY Z$ such that $Y F$ bisects $\angle XY Z$ and $ZF$ bisects $\angle XZY$ . What is the measure of $\angle Y FZ$?

The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.

Given two circles $\omega_1$ and $\omega_2$ where $\omega_2$ is inside $\omega_1$. Show that there exists a point $P$ such that for any line $\ell$ not passing through $P$, if $\ell$ intersects circle $\omega_1$ at $A,B$ and $\ell$ intersects circle $\omega_2$ at $C,D$, where $A,C,D,B$ lie on $\ell$ in this order, then $\angle APC=\angle BPD$.

A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?

Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?

Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

[b]p1.[/b] Calculate $\left( \frac12 + \frac13 + \frac14 \right)^2$. [b]p2.[/b] Find the $2010^{th}$ digit after the decimal point in the expansion of $\frac17$. [b]p3.[/b] If you add $1$ liter of water to a solution consisting of acid and water, the new solutions will contain of $30\%$ water. If you add another $5$ liters of water to the new solution, it will contain $36\frac{4}{11}\%$ water. Find the number of liters of acid in the original solution. [b]p4.[/b] John places $5$ indistinguishable blue marbles and $5$ indistinguishable red marbles into two distinguishable buckets such that each bucket has at least one blue marble and one red marble. How many distinguishable marble distributions are possible after the process is completed? [b]p5.[/b] In quadrilateral $PEAR$, $PE = 21$, $EA = 20$, $AR = 15$, $RE = 25$, and $AP = 29$. Find the area of the quadrilateral. [b]p6.[/b] Four congruent semicircles are drawn within the boundary of a square with side length $1$. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$, where $a$ and $b$ are positive rational numbers. Compute $a + b$. [b]p7.[/b] Let $x$ and $y$ be two numbers satisfying the relations $x\ge 0$, $y\ge 0$, and $3x + 5y = 7$. What is the maximum possible value of $9x^2 + 25y^2$? [b]p8.[/b] In the Senate office in Exie-land, there are $6$ distinguishable senators and $6$ distinguishable interns. Some senators and an equal number of interns will attend a convention. If at least one senator must attend, how many combinations of senators and interns can attend the convention? [b]p9.[/b] Evaluate $(1^2 - 3^2 + 5^2 - 7^2 + 9^2 - ... + 2009^2) -(2^2 - 4^2 + 6^2 - 8^2 + 10^2- ... + 2010^2)$. [b]p10.[/b] Segment $EA$ has length $1$. Region $R$ consists of points $P$ in the plane such that $\angle PEA \ge 120^o$ and $PE <\sqrt3$. If point $X$ is picked randomly from the region$ R$, the probability that $AX <\sqrt3$ can be written in the form $a - \frac{\sqrt{b}}{c\pi}$ , where $a$ is a rational number, $b$ and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find the ordered triple $(a, b, c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABCD$ is a trapezoid with $AB$ parallel to $CD$. The external bisectors of the angles at $ B$ and $C$ intersect at $ P$. The external bisectors of the angles at $ A$ and $D$ intersect at $Q$. Show that the length of $PQ$ is equal to half the perimeter of the trapezoid $ABCD$.

We say that a covering of a $m\times n$ rectangle with dominos has a wall if there exists a horizontal or vertical line that splits the rectangle into two smaller rectangles and doesn't cut any of the dominos. prove that if these three conditions are satisfied: [b]a)[/b] $mn$ is an even number [b]b)[/b] $m\ge 5$ and $n\ge 5$ [b]c)[/b] $(m,n)\neq(6,6)$ then we can cover the rectangle with dominos in such a way that we have no walls. (20 points)

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

Consider a convex qudrilateral $ABCD$ and $M\in (AB),\ N\in (CD)$ such that $\frac{AM}{BM}=\frac{DN}{CN}=k$. Prove that $BC\parallel AD$ if and only if \[MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC\] [i]***[/i]

A triangle with side lengths $2$ and $3$ has an area of $3$. Compute the third side length of the triangle.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that \[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)