$ AL$, $ BM$, and $ CN$ are the medians of $ \triangle ABC$. $ K$ is the intersection of medians. If $ C,K,L,M$ are concyclic and $ AB \equal{} \sqrt 3$, then the median $ CN$ = ? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac {3\sqrt3}{2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$. a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$. b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $a,b,c>0$ and $abc\ge1.$ Prove that a) $\left(a+\frac{1}{a+1}\right)\left(b+\frac{1}{b+1}\right) \left(c+\frac{1}{c+1}\right)\ge\frac{27}{8}.$ b)$27(a^{3}+a^{2}+a+1)(b^{3}+b^{2}+b+1)(c^{3}+c^{2}+c+1)\ge$ $\ge 64(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1).$ [hide="Generalization"]$n^{3}(a^{n}+\ldots+a+1)(b^{n}+\ldots+b+1)(c^{n}+\ldots+c+1)\ge$ $\ge (n+1)^{3}(a^{n-1}+\ldots+a+1)(b^{n-1}+\ldots+b+1)(c^{n-1}+\ldots+c+1),\ n\ge1.$ [/hide]

Find the least odd positive integer that is the middle number of five consecutive integers that are all composite.

Find the largest integer less than $2012$ all of whose divisors have at most two $1$’s in their binary representations. In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

Let $ABC$ be an acute triangle. Consider the points $M, N \in (BC), Q \in (AB), P \in (AC)$ such that the $MNPQ$ is a rectangle. Prove that if the center of the rectangle $MNPQ$ coincides with the center of gravity of the triangle $ABC$, then $AB = AC = 3AP$

The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD $ and $CB = CD.$ Show that $$\angle CAB=\angle CAD.$$

Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. [i]Proposed by Mohammad Gharakhani[/i]

Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$. Prove that $1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4$.

What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. [i]Romania[/i]

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

A token is placed in the leftmost square in a strip of four squares. In each move, you are allowed to move the token left or right along the strip by sliding it a single square, provided that the token stays on the strip. In how many ways can the token be moved so that after exactly $15$ moves, it is in the rightmost square of the strip?

Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?

Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by \[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\] [i]Proposed by Bimit Mandal[/i]

Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]