In triangle $ABC$ $\omega$ is incircle and $\omega_1$,$\omega_2$,$\omega_3$ is tangents to $\omega$ and two sides of $ABC$. $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$. Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$

Let $ABCD$ be a square of center $O$. The parallel to $AD$ through $O$ intersects $AB$ and $CD$ at $M$ and $N$ and a parallel to $AB$ intersects diagonal $AC$ at $P$. Prove that $$OP^4 + \left(\frac{MN}{2} \right)^4 = MP^2 \cdot NP^2$$

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

Find all possibilities: how many acute angles can there be in a convex polygon?

Which statement is not true for at least one prime $p$? $ \textbf{(A)}\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ has a solution, then } \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has a solution.} \\ \\ \textbf{(B)}\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has no solution} \\ \\ \qquad\textbf{(C)}\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ has a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has a solution}. \\ \\ \qquad\textbf{(D)}\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has no solution. } \\ \\ \qquad\textbf{(E)}\ \text{None} $

Evaluate \[\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\sqrt{2055+\ldots}}}}\]

Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that: $ \text{(1)}\sin\in\Lambda $ $ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $ $ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $

Call a number ``Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$.

Find the maximum possible value of the product of distinct positive integers whose sum is $2004$.

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

$a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ and $b_1$, $b_2$, $b_3$, $b_4$, $b_5$, $b_6$, $b_7$ are two permutations of $1, 2, 3, 4, 5, 6, 7$. Show that $|a_1 - b_1|$, $|a_2 - b_2|$, $|a_3 - b_3|$, $|a_4 - b_4|$, $|a_5 - b_5|$, $|a_6 - b_6|$, $|a_7 - b_7|$ are not all different.

Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.

Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number \[ p - \Big\lfloor \frac{p}{q} \Big\rfloor q \] is squarefree (i.e. is not divisible by the square of a prime).

In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that \[ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. \]

Find the number of positive integers less than 100 that are divisors of 300.

\[\int_{-1}^1 \sqrt[3]{x}\log(1+e^x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.

Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$ a) Prove that $P(x)$ has an only integer root. b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$

Let $ABCD$ be a quadrilateral such that $AB = BC = 13$, $CD = DA = 15$ and $AC = 24$. Let the midpoint of $AC$ be $E$. What is the area of the quadrilateral formed by connecting the incenters of $ABE$, $BCE$, $CDE$, and $DAE$?

The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color? [i]Proposed by Eugene Chen [/i]

A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and gets the following answers: $A_1$: On this island there is at least one scoundrel, $A_2$: On this island there are at least two scoundrels, $...$ $A_{n-1}$: On this island there are at least $n-1$ scoundrels, $A_n$: On this island everybody is a scoundrel. Can the journalist decide whether there are more scoundrels or more knights?

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

If the Highest Common Divisor of $ 6432$ and $ 132$ is diminished by $ 8$, it will equal: $ \textbf{(A)}\ \minus{}6 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ \minus{}2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out: (1) If two teams play against each other, there are no more team playing against both of them. (2) If two teams do not play against each other, then there is always exactly two other teams playing against them both. Prove that all teams play the same number of games.

In quadrilateral $ABCD,$ suppose that $\overline{CD}$ is perpendicular to $\overline{BC}$ and $\overline{DA}$. Point $E$ is chosen on segment $\overline{CD}$ such that $\angle AED = \angle BEC$. If $AB = 6$, $AD = 7$, and $\angle ABC = 120^o$ , compute $AE + EB$.