Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

Given is the triangle $ABC$ such that $BC=13, CA=14, AB=15$ Prove that $B$, the incenter $J$ and the midpoints of $AB$ and $BC$ all lie on a circle

Ana, Beatriz, Carlos, Diego and Emilia play a chess tournament. Each player faces each of the other four only once. Each player gets $2$ points if he wins the match, $1$ point if he draws and $0$ point if he loses. At the end of the tournament, it turns out that the scores of the $5$ players are all different. Find the maximum number of ties there could be in the tournament and justify why there could not be a higher number of ties.

A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

A walnut-salesman knows that 20% of the nuts are empty. He has found a test for picking these out. This discards 20% of the nuts. However, when cracking the nuts that were discarded, one fourth of them were not empty after all. What proportion of the nuts that passed the test are then empty? A. 4% B. 6 and 1/4 % C. 8% D. 16% E. None of these

Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$ such that $P_i$ beats $P_{i+1}$ for $i=1, 2, 3, 4$. (We denote $P_5=P_1$).

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3 + 18 + 3$, $18 + 3 + 3$, and $3 + 6 + 3 + 9 + 3$, but not $18 + 6$ or $24$.

Let $ABC$ be an acute-angled scalene triangle and $T$ be a point inside it such that $\angle ATB = \angle BTC = 120^o$. A circle centered at point $E$ passes through the midpoints of the sides of $ABC$. For $B, T, E$ collinear, find angle $ABC$.

Let $a$ and $b$ be two natural numbers whose greatest common divisor is $d$. Prove that exactly $d$ of the numbers $$a, 2a, 3a, ..., (b-1)a, ba$$ is divisible by $b$.

What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 4006$

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

A powderman set a fuse for a blast to take place in $ 30$ seconds. He ran away at a rate of $ 8$ yards per second. Sound travels at the rate of $ 1080$ feet per second. When the powderman heard the blast, he had run approximately: $ \textbf{(A)}\ 200 \text{ yd.} \qquad\textbf{(B)}\ 352 \text{ yd.} \qquad\textbf{(C)}\ 300 \text{ yd.} \qquad\textbf{(D)}\ 245 \text{ yd.} \qquad\textbf{(E)}\ 512 \text{ yd.}$

Let $b\geq 3$ be a positive integer and let $\sigma$ be a nonidentity permutation of the set $\{0,1,\ldots,b-1\}$ such that $\sigma(0)=0.$ The [i]substitution cipher[/i] $C_\sigma$ encrypts every positive integer $n$ by replacing each digit $a$ in the representation of $n$ in base $b$ with $\sigma(a).$ Let $d$ be any positive integer such that $b$ does not divide $d.$ We say that $C_\sigma$ [i]complies[/i] with $d$ if $C_\sigma$ maps every multiple of $d$ onto a multiple of $d,$ and we say that $d$ is [i]cryptic[/i] if there exists some $C_\sigma$ such that $C_\sigma$ complies with $d.$ Let $k$ be any positive integer, and let $p=2^k+1.$ a) Find the greatest power of $2$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. b) Find the greatest power of $p$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. c) Suppose, furthermore, that $p$ is a prime number. Find the greatest cryptic positive integer in base $2p$ and prove that there is only one substitution cipher that complies with it. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Let $a, b$ be two co-prime positive integers. A number is called [i]good [/i] if it can be written in the form $ax + by$ for non-negative integers $x, y$. Define the function $f : Z\to Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is [i]good [/i]if and only if the infinite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.