Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence \[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\] [i]Merlijn Staps[/i]

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?

Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?

Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.

Let \(ABC\) be a triangle with \(AB < AC\) and let \(\omega\) be its circumcircle. Let \(M\) denote the midpoint of side \(BC\) and \(N\) the midpoint of arc \(BC\) of \(\omega\) that contains \(A\). The circumcircle of triangle \(AMN\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Prove that \(BP = CQ\).

Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right. [i]Dorin Popovici[/i]

Which of the following is equal to the product \[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}? \]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$

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?