Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice? $\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

A triangle whose all sides have length not smaller than $1$ is inscribed in a square of side length $1$. Prove that the center of the square lies inside the triangle or on its boundary.

Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.

If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true? I. All shirts in this store are at non-sale prices. II. There is some shirt in this store not on sale. III. No shirt in this store is on sale. IV. Not all shirts in this store are on sale. $ \textbf{(A)}\ \text{II only} \qquad \textbf{(B)}\ \text{IV only} \qquad \textbf{(C)}\ \text{I and III only} \qquad$ $ \textbf{(D)}\ \text{II and IV only} \qquad \textbf{(E)}\ \text{I, II and IV only}$

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.

Find all the pime numbers $(p,q)$ such that : $p^{3}+p=q^{2}+q$

Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

Mary mixes $2$ gallons of a solution that is $40$ percent alcohol with $3$ gallons of a solution that is $60$ percent alcohol. Sandra mixes $4$ gallons of a solution that is $30$ percent alcohol with $\frac{m}{n}$ gallons of a solution that is $80$ percent alcohol, where $m$ and $n$ are relatively prime positive integers. Mary and Sandra end up with solutions that are the same percent alcohol. Find $m + n$.

Solve the equation in nonnegative integers:\\ $2^x=5^y+3$

We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.) \begin{tabular}{l} 1: \textbf{FUNCTION} $f(A)$ \\ 2: $\quad$ \textbf{FOR} $i=1,\ldots, \operatorname{length}(A)-1$: \\ 3: $\quad\quad$ $A[i]\leftarrow A[A[i]]$ \\ 4: $\quad\quad$ $A[0]\leftarrow A[0]-1$ \\ 5: $\quad$ \textbf{RETURN} $A$ \end{tabular} Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?

Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$

Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$, one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.

For an integer $n \ge 0$, let $f(n)$ be the smallest possible value of $ |x + y|$, where $x$ and $y$ are integers such that $3x - 2y = n$. Evaluate $f(0) + f(1) + f(2) +...+ f(2013)$.

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$. In both cases, describe all triples for which equality holds.

Let $ABC$ be a triangle and let the points $D\in BC, E\in AC, F\in AB$, such that \[ \frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB} \] The half-lines $AD, BE,$ and $CF$ intersect the circumcircle of $ABC$ at points $M,N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

Given a figure $B$ in the plane, consider the figures $A$, containing $\mathcal B$, with the property [i]$(P)$: a composition of an odd number of central symmetries with centers in $A$ is also a central symmetry with center in $A$.[/i] The task of this problem is to determine the smallest such figure, denoted by $\mathcal A$, that is contained in every figure $A$. (a) Determine the figure $\mathcal A$ if $B$ consists of: $(1)$ two distinct points $I,J$; $(2)$ three non-collinear points $I,J,K$. (b) Determine $\mathcal A$ if $B$ is a circle (with nonzero radius). (c) Give some examples of figures $B$ whose associated figures $\mathcal A$ are mutually distinct and distinct from the above ones.

prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$

In triangle $ABC$, point$ I$ is incenter , $I_a$ is the $A$-excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$, and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$. Prove that $K$ is the orthocenter of triangle $II_aE$.

Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two. [i]Proposed by David Stoner

The European zoos with at least two types of species are separated in two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A,B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. What is the least $k$ for which we can color the cages in the zoos (each cage only has all animals of one species) such that no zoo has cages of only one color (with every animal across all zoos having to be colored in the same color)? For the maximal value of $k$, find all possibilities (zoos and species), for which this maximum is achieved.

Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$

Let $D$ be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$ be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.

The sequence $(\chi_n) _{n=1}^{\infty}$ is defined as follows \begin{align*} \chi_{n+1}=\chi_n + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*} Prove that none of the terms of this sequence are divisible by $4$