In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$. Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.

A $ 4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? \[ \setlength{\unitlength}{5mm} \begin{picture}(4,4)(0,0) \multiput(0,0)(0,1){5}{\line(1,0){4}} \multiput(0,0)(1,0){5}{\line(0,1){4}} \put(0,3){\makebox(1,1){\footnotesize{1}}} \put(1,3){\makebox(1,1){\footnotesize{2}}} \put(2,3){\makebox(1,1){\footnotesize{3}}} \put(3,3){\makebox(1,1){\footnotesize{4}}} \put(0,2){\makebox(1,1){\footnotesize{8}}} \put(1,2){\makebox(1,1){\footnotesize{9}}} \put(2,2){\makebox(1,1){\footnotesize{10}}} \put(3,2){\makebox(1,1){\footnotesize{11}}} \put(0,1){\makebox(1,1){\footnotesize{15}}} \put(1,1){\makebox(1,1){\footnotesize{16}}} \put(2,1){\makebox(1,1){\footnotesize{17}}} \put(3,1){\makebox(1,1){\footnotesize{18}}} \put(0,0){\makebox(1,1){\footnotesize{22}}} \put(1,0){\makebox(1,1){\footnotesize{23}}} \put(2,0){\makebox(1,1){\footnotesize{24}}} \put(3,0){\makebox(1,1){\footnotesize{25}}} \end{picture} \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Find all triples of reals $(a, b, c)$ such that $$a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.$$

Let $a, b,c$ satises the conditions $\begin{cases} 5 \ge a \ge b \ge c \ge 0 \\ a + b \le 8 \\ a + b + c = 10 \end{cases}$ Prove that $a^2 + b^2 + c^2 \le 38$

Each of the 450 members of a parliament gives a slap in the face to exactly one of his colleagues. Prove that after that they can choose a committee consisting of 150 members, none of whom has been slapped in the face by any other member of the committee. (Folklore)

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.

In a conference $2n$ personalities take apart. Each person has at least $n$ acquaintaces among the others. Prove that it is possible to quarter the participants into two-person rooms, so that each participant would share the room with his/her acquaintace.

On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at $10$ o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at $11$ o'clock, when did he meet the cart?

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.