Consider an angle $ \angle DEF$, and the fixed points $ B$ and $ C$ on the semiline $ (EF$ and the variable point $ A$ on $ (ED$. Determine the position of $ A$ on $ (ED$ such that the sum $ AB\plus{}AC$ is minimum.

Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.

Prove that a prime of the form $2^{2^n}+1$ cannot be the difference of two fifth powers of two positive integers. Pierre.

For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?

In a $7 \times 7$ square table, some of the squares are colored black and the others white, such that each white square is adjacent (along an edge) to an edge of the table or to a black square. Find the minimum number of black squares on the table.

Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

Find the minimum value of \[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\] where $x,y,z>1$ are reals.

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? [asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE);[/asy] $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac7{24} \qquad \textbf{(C)}\ \frac13 \qquad \textbf{(D)}\ \frac38 \qquad \textbf{(E)}\ \frac5{12}$

Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$. Prove that the polynomial \[\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}\] is irreducible in $\mathbb{Q}[X]$. [i]Beniamin Bogosel[/i]

There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.

A jar contains $97$ marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\tfrac{5}{12}.$ After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.

Circle touches parallelogram‘s $ABCD$ borders $AB, BC$ and $CD$ respectively at points $K, L$ and $M$. Perpendicular is drawn from vertex $C$ to $AB$ . Prove, that the line $KL$ divides this perpendicular into two equal parts (with the same length).

It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is positive integer? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 1510 \qquad \textbf{(C)}\ 1511 \qquad \textbf{(D)}\ 1514 \qquad \textbf{(E)}\ \text{None}$

Let $A,B,C,D$ and $E$ be five points lying in this order on a circle, such that $AD=BC$. The lines $AD$ and $BC$ meet at a point $F$. The circumcircles of the triangles $CEF$ and $ABF$ meet again at the point $P$. Prove that the circumcircles of triangles $BDF$ and $BEP$ are tangent to each other.

Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. [i]Ray Li.[/i]