Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.

$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$

A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number? $ \textbf{(A)}\dfrac16\qquad\textbf{(B)}\dfrac5{12}\qquad\textbf{(C)}\dfrac12\qquad\textbf{(D)}\dfrac7{12}\qquad\textbf{(E)}\dfrac56 $

Let $ABC$ be a scalene triangle. Let $I_0=A$ and, for every positive integer $t$, let $I_t$ be the incenter of triangle $I_{t-1}BC$. Suppose that the points $I_0,I_1,I_2,\ldots$ all lie on some hyperbola $\mathcal{H}$ whose asymptotes are lines $\ell_1$ and $\ell_2$. Let the line through $A$ perpendicular to line $BC$ intersect $\ell_1$ and $\ell_2$ at points $P$ and $Q$ respectively. Suppose that $AC^2=\frac{12}{7}AB^2+1$. Then the smallest possible value of the area of quadrilateral $BPCQ$ is $\frac{j\sqrt{k}+l\sqrt{m}}{n}$ for positive integers $j$, $k$, $l$, $m$, and $n$ such that $\gcd(j,l,n)=1$, both $k$ and $m$ are squarefree, and $j>l$. Compute $10000j+1000k+100l+10m+n$. [i]Proposed by Gopal Goel, Luke Robitaille, Ashwin Sah, & Eric Shen[/i]

Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

Let $n$ and $k$ be positive integers. Let $A$ be a set of $2n$ distinct points on the Euclidean plane such that no three points in $A$ are collinear. Some pairs of points in $A$ are linked with a segment so that there are $n^2 + k$ distinct segments on the plane. Prove that there exists at least $\frac{4}{3}k^{3/2}$ distinct triangles on the plane with vertices in $A$ and sides as the aforementioned segments. [i] Proposed by Ho-Chien Chen[/i]

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$ [b](a)[/b] Show that $f$ has exactly one point of discontinuity. [b](b)[/b] Evaluate $f$ at its point of discontinuity.

A $\text{2.25 kg}$ mass undergoes an acceleration as shown below. How much work is done on the mass? [asy] // Code by riben size(350); // Axes draw((0,0)--(12,0),lightgray); draw((0,-3)--(0,5)); // Tick Marks draw((2,0)--(2,-0.2)); label("2",(2,-0.2),S*2); draw((4,0)--(4,-0.2)); label("4",(4,-0.2),S*2); draw((6,0)--(6,-0.2)); label("6",(6,-0.2),S*2); draw((8,0)--(8,-0.2)); label("8",(8,-0.2),S*2); draw((10,0)--(10,-0.2)); label("10",(10,-0.2),S*2); draw((12,0)--(12,-0.2)); label("12",(12,-0.2),S*2); draw((0,-2)--(-0.2,-2)); label("-2",(-0.2,-2),W); draw((0,0)--(-0.2,0),lightgray); label("0",(-0.2,0),W); draw((0,2)--(-0.2,2),lightgray); label("2",(-0.2,2),W); draw((0,4)--(-0.2,4)); label("4",(-0.2,4),W); // Dashed Lines draw((0,-2)--(12,-2),dashed); draw((0,2)--(12,2),dashed+lightgray); draw((0,4)--(12,4),dashed); draw((2,5)--(2,0.2),dashed); draw((4,5)--(4,0.2),dashed); draw((6,5)--(6,0.2),dashed); draw((8,5)--(8,0.2),dashed); draw((10,5)--(10,0.2),dashed); draw((12,5)--(12,0.2),dashed); draw((2,-1)--(2,-3),dashed); draw((4,-1)--(4,-3),dashed); draw((6,-1)--(6,-3),dashed); draw((8,-1)--(8,-3),dashed); draw((10,-1)--(10,-3),dashed); draw((12,-1)--(12,-3),dashed); // Path path A=(0,0)--(2,4)--(4,4)--(6,2)--(8,0)--(10,-2)--(12,0); draw(A,linewidth(1.5)); // Labels label(scale(0.85)*rotate(90)*"Acceleration (m/s/s)",(0,1),W*7); label(scale(0.75)*"Position (m)",(11,0),N); [/asy] (A) $\text{36 J}$ (B) $\text{22 J}$ (C) $\text{5 J}$ (D)$\text{-17 J}$ (E) $\text{-36 J}$

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Show that $$\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}.$$ When does the equality take place?

Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)

For which positive integers $n$ can be covered a ladder like that of the figure (but with $n$ steps instead of $4$) with $n$ squares of integer sides, not necessarily the same size, without these squares overlapping and without standing out from the edge of the figure ?

Let $x_1 \le x_2 \le \ldots \le x_{2n-1}$ be real numbers whose arithmetic mean equals $A$. Prove that $$2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.$$

A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Compute the number of lattice points bounded by the quadrilateral formed by the points $(0, 0)$, $(0, 140)$, $(140, 0)$, and $(100, 100)$ (including the quadrilateral itself). A lattice point on the $xy$ -plane is a point $(x, y)$, where both $x$ and $y$ are integers.

Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$.Is it possible to admit that the initial numbers were a) $\{2, 2, 2\}$? b) $\{3, 3, 3\}$?

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

How many functions $\{f : 1,2, \cdots, 2013\} \rightarrow \{1,2, \cdots, 2013\}$ satisfy $f(j) < f(i) + j - i$ for all integers $i,j$ such that $1 \leq i < j \leq 2013$ ?

Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]