If $ a\plus{}1\equal{}b\plus{}2\equal{}c\plus{}3\equal{}d\plus{}4\equal{}a\plus{}b\plus{}c\plus{}d\plus{}5$, then $ a\plus{}b\plus{}c\plus{}d$ is $ \text{(A)}\ \minus{}5 \qquad \text{(B)}\ \minus{}10/3 \qquad \text{(C)}\ \minus{}7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5$

Let \(n \geq 5\) be integer. The convex polygon \(P = A_{1} A_{2} \ldots A_{n}\) is bicentric, that is, it has an inscribed and circumscribed circle. Set \(A_{i+n}=A_{i}\) to every integer \(i\) (that is, all indices are taken modulo \(n\)). Suppose that for all \(i, 1 \leq i \leq n\), the rays \(A_{i-1} A_{i}\) and \(A_{i+2} A_{i+1}\) meet at the point \(B_{i}\). Let \(\omega_{i}\) be the circumcircle of \(B_{i} A_{i} A_{i+1}\). Prove that there is a circle tangent to all \(n\) circles \(\omega_{i}\), \(1 \leq i \leq n\).

Let $ a,b,c$ be non-zero real numbers satisfying \[ \dfrac{1}{a}\plus{}\dfrac{1}{b}\plus{}\dfrac{1}{c}\equal{}\dfrac{1}{a\plus{}b\plus{}c}.\] Find all integers $ n$ such that \[ \dfrac{1}{a^n}\plus{}\dfrac{1}{b^n}\plus{}\dfrac{1}{c^n}\equal{}\dfrac{1}{a^n\plus{}b^n\plus{}c^n}.\]

Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$. As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.

For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find [b]I.[/b] The maximum and minimum values of $ S(\tau)$. [b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum. [b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.

Solve in equation: $ x^2+y^2+z^2+w^2=3(x+y+z+w) $ where $ x,y,z,w $ are positive integers.

A pair $(a,b)$ of positive integers is good if $\gcd(a,b)=1$ and for each pair of sets $A,B$ of positive integers such that $A,B$ are, respectively, complete residues system modulo $a,b$, there are $x \in A, y \in B$ such that $\gcd(x+y,ab)=1$. For each pair of positive integers $a,k$, let $f(N)$ the number of $b \leq N$ such $b$ has $k$ distinct prime factors and $(a,b)$ is good. Prove that \[\liminf_{n \to \infty} f(n)/\frac{n}{(\log n)^k}\ge e^{k}\]

Does there exist a hexagon that can be divided into four congruent triangles by a straight cut?

A $2\sqrt5$ by $4\sqrt5$ rectangle is rotated by an angle $\theta$ about one of its diagonals. If the total volume swept out by the rotating rectangle is $62\pi$, find the measure of $\theta$ in degrees. [i]Proposed by Connor Gordon[/i]

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.

If $\text{A}$ represents the central atom, in which molecule is the $\text{F-A-F}$ angle the smallest? $ \textbf{(A) } \ce{BF3} \qquad\textbf{(B) }\ce{CF4} \qquad\textbf{(C) }\ce{NF3} \qquad\textbf{(D) }\ce{OF2} \qquad $

Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?

If $\angle A$ is four times $\angle B$, and the complement of $\angle B$ is four times the complement of $\angle A$, then $\angle B=$ $ \textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ} $

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

Point $P$ is located inside an acute-angled triangle $ABC$. $A_1$, $B_1$, $C_1$ are points symmetric to $P$ with respect to the sides of triangle $ABC$. It turned out that the hexagon $AB_1CA_1BC_1$ is inscribed. Prove that $P$ is the Torricelli point of triangle $ABC$.

Given the quadratic trinomial $ f (x) = x ^ 2 + ax + b $ with integer coefficients, satisfying the inequality $ f (x) \geq - {9 \over 10} $ for any $ x $. Prove that $ f (x) \geq - {1 \over 4} $ for any $ x $.

In a triangle $ABC$, let $x=\cos\frac{A-B}{2},y=\cos\frac{B-C}{2},z=\cos\frac{C-A}{2}$. Prove that $$x^4+y^4+z^2\leq 1+2x^2y^2z^2.$$

In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

$S=\{(x,y)|x^2-y^2 \text{is odd},x,y\in\mathbb{R}\},T=\{(x,y)|\sin(2\pi x^2)-\sin(2\pi y^2)=\cos(2\pi x^2)-\cos(2\pi y^2),x,y\in\mathbb{R}\}$, then $\text{(A)}S\subset T\qquad\text{(B)}T\subset S\qquad\text{(C)}S=T\qquad\text{(D)}S\cap T=\varnothing$

Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $AB, BC$ and $CA $ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A$DIF$ and $BDIE$. Let $J_1J_2$ intersect $AB$ at point $M$. Prove that $CD$ is perpendicular to $IM$.

A cube has side length $5$. Let $S$ be its surface area and $V$ its volume. Find $\frac{S^3}{V^2}$ .

In Wonderland, the towns are connected by roads, and whenever there is a direct road between two towns there is also a route between these two towns that does not use that road. (There is at most one direct road between any two towns.) The Queen of Hearts ordered the Spades to provide a list of all ”even” subsystems of the system of roads, that is, systems formed by subsets of the set of roads, where each town is connected to an even number of roads (possibly none). For each such subsystem they should list its roads. If there are totally $n$ roads in Wonderland and $x$ subsystems on the Spades’ list, what is the number of roads on their list when each road is counted as many times as it is listed?

For any positive integer $n$, let $a_n$ be the number of pairs $(x,y)$ of integers satisfying $|x^2-y^2| = n$. (a) Find $a_{1432}$ and $a_{1433}$. (b) Find $a_n$ .