Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.

The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by \[b_n=\sum_{k=0}^n \binom nk a_k.\] Find the coefficients $\alpha,\beta$  so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.

Two circles $\omega_1,\omega_2$ intersect at $A,B$. An arbitrary line through $B$ meets $\omega_1,\omega_2$ at $C,D$ respectively. The points $E,F$ are chosen on $\omega_1,\omega_2$ respectively so that $CE=CB,\ BD=DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega_2$ at $Q$. Prove that $A,P,Q$ are collinear. [i]Proposed by Iman Maghsoudi[/i]

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\] [i]Proposed by snap7822[/i]

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

From a point $K$ of a circle, a chord $KA$ (arc $AK$ is greather than $90^{o}$) and a tangent $l$ are drawn. The line that passes through the center of the circle and that is perpendicular to the radius $OA$, intersects $KA$ at $B$ and $l$ at $C$. Show that $KC = BC$.

The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below. [center][img]https://snag.gy/LGimC8.jpg[/img][/center]

Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.

Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$? $\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

Let be two natural numbers $ n $ and $ a. $ [b]a)[/b] Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality. $$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$ [b]b)[/b] Show that there exist only finitely such $ n\text{-tuplets} . $

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions: [b](i)[/b] if $f \in A$ and $\deg( f ) \leq 1$, then $f(x) = x - 1$; [b](ii)[/b] if $f \in A$ and $\deg( f ) \geq 2$, then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$; [b](iii)[/b] for every $g, h \in A$, both $x^{2+\deg(g)} + xg(x) -1$ and $x^{1+\deg(g)}g(x) + h(x)$ belong to $A.$ Let $R_n(f)$ be the remainder of the Euclidean division of the polynomial $f(x)$ by $x^n$. Prove that for all $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$, and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$.

For all $n$, $t_{n+1} = 2(t_n)^2 - 1$. Prove that gcd $(t_n,t_m) = 1$ if $n \ne m$.

A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.

The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy \[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\] for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.

Let $ABCD$ be a quadrilateral with side lengths $AB = 2$, $BC = 3$, $CD = 5$, and $DA = 4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$?

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $