If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? $\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}7 \qquad \textbf{(C)}\hspace{.05in}8 \qquad \textbf{(D)}\hspace{.05in}9 \qquad \textbf{(E)}\hspace{.05in}10 $

Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O$, and that the ratio of the volume of $O$ to that of $C$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

Evaluate the following limit $$\lim_{x\to 0} (1 + 2x + 3x^2 + 4x^3 +...)^{1/x}$$

An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?

In a convex qualrilateral $ABCD$, let $P$ be the intersection point of $AD$ and $BC$. Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$,respectively. Let $O$ be the circumcenter of $PAB$, and $H$ the orthocenter of $PDC$. Show that the circumcircles of triangles $AI_1B$ and $DHC$ are tangent together if and only if the circumcircles of triangles $AOB$ and $DI_2C$ are tangent together. Proposed by Hooman Fattahimoghaddam

Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?

Given three distinct digits a, b and c, it is possible, by choosing two digits at a time, to form six two-digit numbers. Determine all possible sets {a, b, c} for which the sum of the six two-digits numbers is 484.

Let $ABCD$ be a convex quadrilateral. Ray $AD$ meets ray $BC$ at $P$. Let $O,O'$ be the circumcenters of triangles $PCD, PAB$, respectively, $H,H'$ be the orthocenters of triangles $PCD, PAB$, respectively. Prove that circumcircle of triangle $DOC$ is tangent to circumcircle of triangle $AO'B$ if and only if circumcircle of triangle $DHC$ is tangent to circumcircle of triangle $AH'B$.

The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $$\log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015,$$ find the number of possible ordered pairs $(a,r)$.

Circle $(O)$ and its chord $BC$ are given. Point $A$ moves on the major arc $BC$. $AL$ is the angle bisector in a triangle $ABC$. Show that the disctance from the circumcenter of triangle $AOL$ to the line $BC$ does not depend on the position of point $A$.

Find the numerator of \[\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\] when reduced.

Let $n > 10$ be an odd integer. Determine the number of ways to place the numbers $1, 2, \ldots , n$ around a circle so that each number in the circle divides the sum its two neighbors. (Two configurations such that one can be obtained from the other per rotation are to be counted only once.)

The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$

Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.

Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$.

Anna and Ben decided to visit Archipelago with $2009$ islands. Some pairs of islands are connected by boats which run both ways. Anna and Ben are playing during the trip: Anna chooses the first island on which they arrive by plane. Then Ben chooses the next island which they could visit. Thereafter, the two take turns choosing an island which they have not yet visited. When they arrive at an island which is connected only to islands they had already visited, whoever's turn to choose next would be the loser. Prove that Anna could always win, regardless of the way Ben played and regardless of the way the islands were connected. [i](12 points for Juniors and 10 points for Seniors)[/i]

A circle is located on the plane. What is the smallest number of lines you need to draw so that, symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), it could cover any given point of the plane?

Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? [i]Proposed by Nathan Xiong[/i]

For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that \[b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.\]

Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.