Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list. a) Prove that we can always find a good list with 12 elements, whatever the lists are; b) Prove that students can make lists such that no shorter list is good.

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4. [i]Laurentiu Panaitopol[/i]

Determine all solutions to the system of equations: \[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\] [This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.

Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.

Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$ [i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]

Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$.

Find all natural numbers $n$ such that $n$, $n^2+10$, $n^2-2$, $n^3+6$, and $n^5+36$ are all prime numbers.

Among $239$ coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least $x$ times as large as the shorter piece is $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{2}{x}\qquad\textbf{(C) }\frac{1}{x+1}\qquad\textbf{(D) }\frac{1}{x}\qquad \textbf{(E) }\frac{2}{x+1}$

Given a rectangle $ABCD$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively so that the area of triangles $ABE$, $ECF$, $FDA$ is equal to $1$. Determine the area of triangle $AEF$.

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Let $p$ be a prime number greater than 10. Prove that there exist positive integers $m$ and $n$ such that $m+n < p$ and $5^m 7^n-1$ is divisible by $p$.

Let $ABC$ be an acute angled triangle and let $P, Q$ be points on $AB, AC$ respectively, such that $PQ$ is parallel to $BC$. Points $X, Y$ are given on line segments $BQ, CP$ respectively, such that $\angle AXP = \angle XCB$ and $\angle AYQ = \angle YBC$. Prove that $AX = AY$. [i]Proposed by Ervin Maci$\acute{c},$ Bosnia and Herzegovina[/i]

Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment). For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given? *Here, the [i]centroid[/i] of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$. [i]Proposed by Holden Mui and Carl Schildkraut[/i]

a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that \[a^2+b^2+c^2+abc<8.\] b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have \[a^2+b^2+c^2+abc<d \quad\] where $a,b$ and $c$ are the side lengths of the triangle?

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$, and let $P$ be the midpoint of the minor arc $BC$ of $\Gamma$. Let $AP$ and $BC$ meet at $D$, and let $M$ be the midpoint of $AB$. Also, let $E$ be the point such that $AE\perp AB$ and $BE\perp MP$. Prove that $AE=DE$.

A regular $2015$-simplex $\mathcal P$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal P$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$. [i] Proposed by James Lin [/i]

A triangle $ABC$ is given in which $\angle BAC = 40^o$. and $\angle ABC = 20^o$. Find the length of the angle bisector drawn from the vertex $C$, if it is known that the sides $AB$ and $BC$ differ by $4$ centimeters.

In the $43$ dimension Euclidean space the convex hull of finite set $S$ contains polyhedron $P$. We know that $P$ has $47$ vertices. Prove that it is possible to choose at most $2021$ points in $S$ such that the convex hull of these points also contain $P$, and this is sharp.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.