Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by $l$, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of $l$ and t. Show that CE and t are perpendicular to each other.

The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that $$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$

How many primes $p$ are there such that $39p + 1$ is a perfect square? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 3 \qquad\textbf{e)}\ \text{None of above} $

Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting $32$ triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.

Let m be a positive interger, let $p$ be a prime, let $a_1=8p^m$, and let $a_n=(n+1)^{\frac{a_{n-1}}{n}}$, $n=2,3...$. Determine the primes $p$ for which the products $a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})$, $n=1,2,3...$ are all integral.

An infinite sequence of positive real numbers is defined by $a_0=1$ and $a_{n+2}=6a_n-a_{n+1}$ for $n=0,1,2,\cdots$. Find the possible value(s) of $a_{2007}$.

There are $3$ classes with $n$ students in each class, and the heights of all $3n$ students are pairwise distinct. Partition the students into groups of $3$ such that in each group, there is one student from each class. In each group, call the tallest student the [i]tall guy[/i]. Suppose that for any partition of the students, there are at least 10 tall guys in each class, prove that the minimum value of $n$ is $40$.

For $n \ge 2,$ let $\omega(n)$ denote the number of distinct prime factors of $n.$ We set $\omega(1) = 0.$ Compute the absolute value of $$\sum_{n=1}^{160} (-1)^{\omega(n)} \left\lfloor \frac{160}{n} \right\rfloor.$$

Determine the remainder when $$2^{\frac {1 \cdot 2}{2}} + 2^{\frac {2 \cdot 3}{2}}+ ...+ 2^{\frac {2011 \cdot 2012}{2}}$$ is divided by $7$.

Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$

Two players are writting in turn natural numbers not exceeding $p$. The rules forbid to write the divisors of the numbers already having been written. Those who cannot make his move looses. a) Who, and how, can win if $p=10$? b) Who wins if $p=1000$?

In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10.$ Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG.$ [asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines. Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?

On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.