Prove, that for every natural $N$ exists $k$, such that $N=a_02^0+a_12^1+...+a_k2^k$, where $a_0,a_1,...a_k$ are $1$ or $2$

The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$ A. Voidelevich

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$

A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?

Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.

Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.

Consider any permutation $\sigma$ of $\{0,1,2,\dots,9\}$ and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let $M(\sigma)$ be the largest of these 8 sums. (For example, for the permutation $\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3)$ we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and $M(\sigma) = 20$.) (a) Find a permutation $\sigma_1$ such that $M(\sigma_1) = 13$. (b) Does there exist a permutation $\sigma_2$ such that $M(\sigma_2) = 12$?

In an acute triangle $ABC$, a point $P$ is taken on the $A$-altitude. Lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at points $D$ and $E$, respectively. Tangents drawn from points $D$ and $E$ to the circumcircle of triangle $BPC$ are tangent to it at points $K$ and $L$, respectively, which are in the interior of triangle $ABC$. Line $KD$ intersects the circumcircle of triangle $AKC$ at point $M$ for the second time, and line $LE$ intersects the circumcircle of triangle $ALB$ at point $N$ for the second time. Prove that\[ \frac{KD}{MD}=\frac{LE}{NE} \iff \text{Point P is the orthocenter of triangle ABC}\]

There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed: -the first card (from the top) is put in the bottom of the deck. -the second card (from the top) is taken away of the deck. -the third card (from the top) is put in the bottom of the deck. -the fourth card (from the top) is taken away of the deck. - ... The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.

Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.

Consider the sequence $z_n = (1+i)(2+i)...(n+i)$. Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.