A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

Let be an $ n\times n $ positive-definite symmetric real matrix $ A. $ Prove the following equality. $$ \tiny\int_{\mathbb{R}^n} \exp\left( -\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}^\intercal A\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\right) dx_1dx_2\cdots dx_n=\normalsize\frac{\pi^{n/2}}{\sqrt{\det A} } $$

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? [asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\displaystyle\frac12$",(0.25,0),S); label("$\displaystyle\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy] $ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? $ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $

A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)

Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.

Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

Can a set of six numbers $\left\{a, b,c, \frac{a^2}{b} , \frac{b^2}{c} , \frac{c^2}{a} \right\}$ , where $a, b, c$ positive numbers, turn out to be exactly exactly three different numbers?

A mathematics teacher writes a quadratic equation on the blackboard of the form $$x^2+mx \star n = 0$$, with $m$ and $n$ integers. The sign of $n$ is blurred. Even so, Claudia solves it and obtains integer solutions, one of which is $2011$. Find all possible values of $m$ and $n$.

Let $P$ be a set of $n \ge 2$ distinct points in the plane, which does not contain any triplet of aligned points. Let $S$ be the set of all segments whose endpoints are points of $P$. Given two segments $s_1, s_2 \in S$, we write $s_1 \otimes s_2$ if the intersection of $s_1$ with $s_2$ is a point other than the endpoints of $s_1$ and $s_2$. Prove that there exists a segment $s_0 \in S$ such that the set $\{s \in S | s_0 \otimes s\}$ has at least $\frac{1}{15}\dbinom{n-2}{2}$ elements

Can we build parallelepiped $6 \times 7 \times 7$ from $1 \times 1 \times 2$ bricks, such that we have same amount bricks of one of $3$ directions ?

For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition: For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.

Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime.

Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights. Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.

The chord $[CD]$ is parallel to the diameter $[AB]$ of a circle with center $O$. The tangent line at $A$ meet $BC$ and $BD$ at $E$ and $F$. If $|AB|=10$, calculate $|AE|\cdot |AF|$.

Find all prime numbers $p$ for which there exist quadratic trinomials $P(x)$ and $Q(x)$ with integer coefficients, both with leading coefficients equal to $1$, such that the coefficients of $x^0$, $x^1$, $x^2$, and $x^3$ in the expanded form of the product $P(x)Q(x)$ are congruent modulo $p$ to $4$, $0$, $(-16)$, and $0$, respectively.

In triangle $ABC$, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD, DE $ and $EC$ are $1, 3 $ and $5$ respectively. Find the length of $AC$.

Mummy-trolley huts are located on a straight line at points with coordinates $x_1, x_2,...., x_n$. In this village are going to build $3$ stores $A, B$ and $C$, of which will be brought every day to all Moomin-trolls chocolates, bread and water. For the delivery of chocolate, the store takes the distance from the store to the hut, raised to the square; for bread delivery , take the distance from the store to the hut; for water delivery take distance $1$, if the distance is greater than $1$ km, but do not take anything otherwise. a) Where to build each of the stores so that the total cost of all Moomin-trolls for delivery wasthe smallest? b) Where to place the TV tower, if the fee for each Moomin-troll is the maximum distance from the TV tower to the farthest hut from it? c) How will the answer change if the Moomin-troll huts are not located in a straight line, and on the plane? [hide=original wording] На прямiй розташованi хатинки Мумi-тролей в точках з координатами x1, x2, . . . , xn. В цьому селi бираються побудувати 3 магазина A, B та C, з яких будуть кожен день привозити всiм Мумi-тролям шоколадки, хлiб та воду. За доставку шоколадки мага- зин бере вiдстань вiд магазину до хатинки, пiднесену до квадрату; за доставку хлiба – вiдстань вiд магазину до хатинки; за доставку води беруть 1, якщо вiдстань бiльша 1 км, та нiчого не беруть в супротивному випадку. 1. Де побудувати кожний з магазинiв, щоб загальнi витрати всiх Мумi-тролей на доставку були найменшими? 2. Де розташувати телевежу, якщо плата для кожного Мумi-троля – максимальна вiдстань вiд телевежi до самої вiддаленої вiд неї хатинки? 3. Як змiниться вiдповiдь, якщо хатинки Мумi-тролей розташованi не на прямiй, а на площинi?[/hide]

Initially, a positive integer is written on the blackboard. Every second, one adds to the number on the board the product of all its nonzero digits, writes down the results on the board, and erases the previous number. Prove that there exists a positive integer which will be added inifinitely many times.

Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.

In a triangle $ ABC$, the bisectors of the angles at $ B$ and $ C$ meet the opposite sides $ B_1$ and $ C_1$, respectively. Let $ T$ be the midpoint $ AB_1$. Lines $ BT$ and $ B_1C_1$ meet at $ E$ and lines $ AB$ and $ CE$ meet at $ L$. Prove that the lines $ TL$ and $ B_1C_1$ have a point in common.

Prove that in each polyhedron there exist two faces with the same number of edges.