Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.

On each side of a triangle, $5$ points are chosen (other than the vertices of the triangle) and these $15$ points are colored red. How many ways are there to choose four red points such that they form the vertices of a quadrilateral?

Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.

Bored on their trip home, Joshua and Alexis decide to keep a tally of license plates they see in the other lanes: Joshua watches cars going the other way, and Alexis watches cars in the next lane. After a few minutes, Wendy counts up the tallies and declares, "Joshua has counted $2008$ license plates, and there are $17$ license plate designs he's seen exactly $17$ times, but of Alexis's $2009$ license plates, there's none she's seen exactly $18$ times. Clearly, $17$ is the specialist number." Michael, suspicious, pulls out the $\textit{Almanac of American License Plates}$ and notes, "According to confirmed demographic statistics, you'd only expect those numbers to be $5.4$ and $4.9$, respectively. But the $17^\text{th}$ state is weird: Joshua saw exactly $17$ of its license plates, which isn't what we'd expect." Alexis asks, "How many Ohioan license plates did we expect to see?" and reaches for the $\textit{Almanac}$ to find out, but Michael snatches it away and says, "I'm not telling." Alexis, disappointed, says, "I suppose that $17$ is my best guess," feeling that the answer must be pretty close to $17$. Wendy smiles. "You can do better than that, actually. Given what Michael said and that we saw $17$ Ohioan license plates, we'd actually expect there to have been $\tfrac ab$ less than $17$." Help Alexis. If $\tfrac ab$ is in lowest terms, find the product $ab$.

A pond has $2025$ lily pads arranged in a circle. Two frogs, Alice and Bob, begin on different lily pads. A frog jump is a jump which travels $2$, $3$, or $5$ positions clockwise. Alice and Bob each make a series of frog jumps, and each frog ends on the same lily pad that it started from. Given that each lily pad is the destination of exactly one jump, prove that each frog completes exactly two laps around the pond (i.e. travels $4050$ positions in total). [i]Linus Tang[/i]

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?

Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$ for all positive integers $n$.

Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.

Triangle $ABC$ has incircle $O$ that is tangent to $AC$ at $D$. Let $M$ be the midpoint of $AC$. $E$ lies on $BC$ so that line $AE$ is perpendicular to $BO$ extended. If $AC = 2013$, $AB = 2014$, $DM = 249$, find $CE$.

The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula if \[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \] Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]

Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces. [i]Proposed by Kei Irei, Japan[/i]

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? $ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

Solve the equation $$arc \sin (\sin x) + arc \cos (\cos x)=0$$

Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$. Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$. [hide="Comment"]In the contest, the problem was given with a) and b): a) Prove the above for $n=2$; b) Prove the above for $n\geq 3$ as well.[/hide]

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

Triangles $ ABC$ and $ A_1B_1C_1$ have the same area. Using compass and ruler, can we always construct triangle $ A_2B_2C_2$ equal to triangle $ A_1B_1C_1$ so that the lines $ AA_2$, $ BB_2$, and $ CC_2$ are parallel?

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.