Consider a circle of radius $6$. Let $B,C,D$ and $E$ be points on the circle such that $BD$ and $CE$, when extended, intersect at $A$. If $AD$ and $AE$ have length $5$ and $4$ respectively, and $DBC$ is a right angle, then show that the length of $BC$ is $\frac{12+9\sqrt{15}}{5}$.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

The value of the expression $ \frac{(304)^5}{(29.7)(399)^4}$ is closest to \[ \textbf{(A)}\ .003 \qquad \textbf{(B)}\ .03 \qquad \textbf{(C)}\ .3 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 30 \]

In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $. [hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\prod_{j=1}^n (x-j)^j, $$ where $ n $ is a natural number.

A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list? ${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D)}}\ 33\qquad\textbf{(E)}\ 35 $

How many letters in the word UNCOPYRIGHTABLE have at least one line of symmetry?

Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: $ \textbf{(A)}\ 1080\qquad \textbf{(B)}\ 900\qquad \textbf{(C)}\ 720\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 360$

What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$

Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.

Let $\Gamma$ be a semicircle with center $O$ and diameter $AB$. $D$ is the midpoint of arc $AB$. On the ray $OD$, we take $E$ such that $OE = BD$. $BE$ intersects the semicircle at $F$ and $ P$ is the point on $AB$ such that $FP$ is perpendicular to $AB$. Prove that $BP=\frac13 AB$.

A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.

$ A$ can do a piece of work in $ 9$ days. $ B$ is $ 50\%$ more efficient than $ A$. The number of days it takes $ B$ to do the same piece of work is: $ \textbf{(A)}\ 13\frac {1}{2} \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these answers}$

What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram. [i]Proposed by Iman Maghsoudi[/i]

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance? $\textbf{(A) }7.2\hspace{14em}\textbf{(B) }14.4\hspace{14em}\textbf{(C) }36$ $\textbf{(D) }10\hspace{14.3em}\textbf{(E) }12\hspace{14.8em}\textbf{(F) }2.4$ $\textbf{(G) }25.2\hspace{13.5em}\textbf{(H) }123456789$

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$. [i]Proposed by Yang Liu[/i]