The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

$20\%$ of $10$ is exactly $5\%$ of what number? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }40\qquad\textbf{(D) }50\qquad\textbf{(E) }60$

X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?

$\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$ $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{2}{3} \qquad \text{(C)}\ \dfrac{3}{2} \qquad \text{(D)}\ \dfrac{17}{3} \qquad \text{(E)}\ \dfrac{34}{3}$

Austin is at the Lincoln Airport. He wants to take $5$ successive flights whose destinations are randomly chosen among Indianapolis, Jackson, Kansas City, Lincoln, and Milwaukee. The origin and destination of each flight may not be the same city, but Austin must arrive back at Lincoln on the last of his flights. Compute the probability that the cities Austin arrives at are all distinct.

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent. [i]Pablo Jaén, Panama[/i]

Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that: \[\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}\]

Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$? $\textbf{(A) }{-}7\qquad\textbf{(B) }{-}5\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$

Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$

There are $2^n$ airports, numbered with binary strings of length $n{}$. Any two stations whose numbers differ in exactly one digit are connected by a flight that has a price (which is the same in both directions). The sum of the prices of all $n{}$ flights leaving any station does not exceed 1. Prove that one can travel between any two airports by paying no more than 1.

Given an $n \times n$ ($n \ge 3$) square table with one of the following unit vectors $\uparrow, \downarrow, \leftarrow, \rightarrow$ in any its cell (the vectors are parallel to the sides and the middles of them coincide with the centers of the cells). Per move a beetle creeps from one cell to another in accordance with the vector’s direction. If the beetle starts from any cell, then it comes back to this cell after some number of moves. The vectors are directed so that they do not allow the beetle to leave the table. Is it possible that the sum of all vectors at any row (except for the first one and the last one) is equal to the vector that is parallel to this row, and the sum of all vectors at any column (except for the first one and the last one) is equal to the vector that is parallel to this column ? (D. Dudko)

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$.

Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.