Let $FELDSPAR$ be a regular octagon, and let $I$ be a point in its interior such that $\angle FIL = \angle LID =$ $\angle DIS$ $=\angle SIA.$ Compute $\angle IAR$ in degrees.

Determine all $a\in\mathbb R$ such that the inequality \[x^4+x^3-2(a+1)x^2-ax+a^2<0\] has at least one real solution $x.$

Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.

Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.

Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that $$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$ Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.

Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that \[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\] Prove that \[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\] [i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]

Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$. (i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$. (ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ . (iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

The following geometry is embedded in the Cartesian plane. [b]a)[/b] Prove that any line that passes through at least two lattice points, passes through at least three lattice points. [b]b)[/b] Find a point on the plane which doesn't lie on any line that passes through at least two lattice points. [b]c)[/b] Show that any point with rational coordinates lie on a line that passes through two lattice points. [i]Lavinia Savu[/i]

We consider the following system with $q=2p$: \[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\] in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties: [b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$ [b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$ [b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

Let $ ABC$ be a triangle with $ BC \equal{} 2007$, $ CA \equal{} 2008$, $ AB \equal{} 2009$. Let $ \omega$ be an excircle of $ ABC$ that touches the line segment $ BC$ at $ D$, and touches extensions of lines $ AC$ and $ AB$ at $ E$ and $ F$, respectively (so that $ C$ lies on segment $ AE$ and $ B$ lies on segment $ AF$). Let $ O$ be the center of $ \omega$. Let $ \ell$ be the line through $ O$ perpendicular to $ AD$. Let $ \ell$ meet line $ EF$ at $ G$. Compute the length $ DG$.

Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that \[ \Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, . \]

On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it? [asy] size(450); defaultpen(linewidth(0.8)); path[] p={origin--(8,8)--(14,8), (0,10)--(4,10)--(14,0), origin--(14,14), (0,14)--(14,14), origin--(7,7)--(14,0)}; int i; for(i=0; i<5; i=i+1) { draw(shift(21i,0)*((0,16)--origin--(14,0))); draw(shift(21i,0)*(p[i])); label("Time", (7+21i,0), S); label(rotate(90)*"Volume", (21i,8), W); } label("$A$", (0*21 + 7,-5), S); label("$B$", (1*21 + 7,-5), S); label("$C$", (2*21 + 7,-5), S); label("$D$", (3*21 + 7,-5), S); label("$E$", (4*21 + 7,-5), S); [/asy] $\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$

The permutations of $OLYMPIAD$ are arranged in lexicographical order, with $ADILMOPY$ being arrangement 1 and its reverse being arrangement $40320$. Yu Semo and Yu Sejmo both choose a uniformly random arrangement. The immature Yu Sejmo exclaims, ``My fourth letter is $L$!" while Yu Semo remains silent. Given this information, let $E_1$ be the expected arrangement number of Yu Semo and $E_2$ be the expected arrangement number of Yu Sejmo. Compute $E_2 - E_1$.