We call a positive integer [i]lively[/i] if it has a prime divisor greater than $10^{10^{100}}$. Prove that if $S$ is an infinite set of lively positive integers, then it has an infinite subset $T$ with the property that the sum of the elements in any finite nonempty subset of $T$ is a lively number.

Find all pairs of real numbers $(x, y) $ such that: a) $x\ge y\ge1$ b) $2x^2-xy-5x +y + 4 = 0 $

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?

Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?

Three primes $p,q,$ and $r$ satisfy $p+q = r$ and $1 < p < q$. Then $p$ equals $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 17 $

Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.

Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$, one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod\; c)$ has integral solutions.

There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.

Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

If $x, y, z$ satisfy $x+y+z = 12, \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$ and $x^3+y^3+z^3 = -480,$ find $$x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2.$$ [i]Proposed by Mahith Gottipati[/i]

Let $f(x)$ and $g(x)$ be two polynomials with real coefficients such that for infinitely many rational values of $x$, the fraction $\frac{f(x)}{g(x)}$ is rational. Prove that $\frac{f(x)}{g(x)}$ can be written as the ratio of two polynomials with rational coefficients.

A tower of height $h$ is a stack of contiguous rows of squares of height $h$ such that [list] [*] the bottom row of the tower has $h$ squares, [*] each row above the bottom row has one fewer square than the row below it, and within each row the squares are contiguous, [*] the squares in any given row all lie directly above a square in the row below. [/list] A tower is called balanced if when the squares of the tower are colored black and white in a checkerboard fashion, the number of black squares is equal to the number of white squares. For example, the figure above shows a tower of height 5 that is not balanced, since there are 7 white squares and 8 black squares. How many balanced towers are there of height 2016?

John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$

If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. First $B$ goes more slowly than $A$ until they meet then by doubling his rate he next meets $A$ at the starting point. Let $d$ m be the distance travelled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.

$\Delta ABC$ is right-angled at $B$, with $AB=1$ and $BC=3$. $E$ is the foot of perpendicular from $B$ to $AC$. $BA$ and $BE$ are produced to $D$ and $F$ respectively such that $D$, $F$, $C$ are collinear and $\angle DAF=\angle BAC$. Find the length of $AD$.

The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

Paul and his pet octahedron like to play games together. For this game, the octahedron randomly draws an arrow on each of its faces pointing to one of its three edges. Paul then randomly chooses a face and progresses from face to adjacent face, as determined by the arrows on each face, and he wins if he reaches every face of the octahedron. What is the probability that Paul wins?

\[\int_0^1 (x^6+6x^5+15x^4+15x^2+6x+1)\mathrm dx\] [i]Proposed by Robert Trosten[/i]