How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? $\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{None}$

Prove for real $x_1$, $x_2$, ....., $x_n$, $0 < x_k \leq {1\over 2}$, the inequality \[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]

In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$

Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$

In rectangle $ABCD, AB= 5$ and $BC = 3$. Points $F$ and $G$ are on line segment $CD$ so that $DF = 1$ and $GC = 2$. Lines $AF$ and $BG$ intersect at $E$. What is the area of $\vartriangle AEB$?

Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$. Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$. Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.

Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$ and $P$ its intersection of diagonals. Denote by $O_1$, $O_2$ the circumcenters of triangles $ABP$, $CDP$, respectively. Prove that $O_1BCO_2$ is a parallelogram. [i] (Patrik Bak)[/i]

The circumference of a circle is arbitrarily divided into four arcs. The midpoints of the arcs are connected by segments. Show that two of these segments are perpendicular.

An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of $1$ unit either up or to the right. A lattice point $(x, y)$ with $0 \le x, y \le 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0, 0)$ to$ (5,5)$ not passing through $(x, y)$.

A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$. [asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]

A binary sequence is constructed as follows. If the sum of the digits of the positive integer $k$ is even, the $k$-th term of the sequence is $0$. Otherwise, it is $1$. Prove that this sequence is not periodic.

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively. (a) Prove that line $OH$ intersects both segments $AB$ and $AC$. (b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$. a) Prove that $IK$ and $OE$ are parallel. b) Prove that $PK$ is perpendicular to $IQ$.

In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

With positive $ a,b,c, $ prove: $$ \frac{a}{8a^2+5b^2+3c^2} +\frac{b}{8b^2+5c^2+3a^2} +\frac{c}{8c^2+5a^2+3b^2}\le\frac{1}{16}\left( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} \right) $$ [i]Titu Zvonaru[/i]

Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.

Consider three pairwise intersecting circles $\omega_1$, $\omega_2$ and $\omega_3$. Let their three common chords intersect at point $R$. We denote by $O_1$ the center of the circumcircle of a triangle formed by some triple common points of $\omega_1$ & $\omega_2$, $\omega_2$ & $\omega_3$ and $\omega_3$ & $\omega_1$. and we denote by $O_2$ the center of the circumcircle of the triangle formed by the second intersection points of the same pairs of circles. Prove that points $R$, $O_1$ and $O_2$ are collinear.

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$. [i]Chen Yonggao[/i]

For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$. [i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]

Let $ABCDE$ be a regular pentagon with center $M$. A point $P\neq M$ is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$. Prove that $AR$ and $QR$ are of the same length.