Which of the following numbers is a perfect square? $ \textbf{(A)}\ 98!\cdot 99!\qquad \textbf{(B)}\ 98!\cdot 100!\qquad \textbf{(C)}\ 99!\cdot 100!\qquad \textbf{(D)}\ 99!\cdot 101!\qquad \textbf{(E)}\ 100!\cdot 101!$

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.

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$ $\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.

The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.

For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence? $ \textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4,030,056 \qquad \textbf{(E)}\ 4,032,064$

Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$. Show that the rays $[AX$ and $[BY$ intersect on line $CM$.

Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define \[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\] Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum \[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\] $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Let $ G$ be finite group and $ \mathcal{K}$ a conjugacy class of $ G$ that generates $ G$. Prove that the following two statements are equivalent: (1) There exists a positive integer $ m$ such that every element of $ G$ can be written as a product of $ m$ (not necessarily distinct) elements of $ \mathcal{K}$. (2) $ G$ is equal to its own commutator subgroup. [i]J. Denes[/i]

The number of intengral points $(x,y)$ that fit $(|x|-1)^2+(|y|-1)^2<2$ is $\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25$

Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.

Let a, b, c be positive integers. Prove that the inequality \[ (x\minus{}y)^a(x\minus{}z)^b(y\minus{}z)^c \ge 0 \] holds for all reals x, y, z if and only if a, b, c are even.

Let n be an integer such that if d | n then d + 1 | n + 1. Show that n is a prime number.

Set $M=\{1,2,\cdots,1995\}$. $A$ is a subset of $M$ such that $\forall x\in A,15x\not\in A$. Then the maximum $|A|$ is________.

Equilateral triangle $OAB$ of side length $1$ lies in the $xy$-plane ($O$ is the origin). Let $\ell, m$ be the vertical lines passing through $A,B$, respectively. Let $P,Q$ be on $\ell, m$ respectively such that the ratio $\overline{OP} : \overline{OQ} : \overline{PQ} = 3 : 3 : 5$. Let $Q = (x, y, z)$. If $z^2 = \frac{p}{q}$ . where $p, q$ are relatively prime positive integers, find $p + q$.

Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which $f(k+1)>f(f(k)) \quad \forall k \geq 1$.

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{mn}-F_{n+1}^{m}+F_{n-1}^{m}$ is divisible by $F_{n}^{3}$ for all $m \ge 1$ and $n>1$.

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

More than three quarters of the circumference of a circle is colored black. Prove that there exists a rectangle such that all of its vertices are black. Actually the result holds if "three quarters" is replaced by "one half"...

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that $$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$

Let $ABC$, $AB<AC$ be an acute triangle inscribed in circle $\Gamma$ with center $O$. The altitude from $A$ to $BC$ intersects $\Gamma$ again at $A_1$. $OA_1$ intersects $BC$ at $A_2$ Similarly define $B_1$, $B_2$, $C_1$, and $C_2$. Then $B_2C_2=2\sqrt{2}$. If $B_2C_2$ intersects $AA_2$ at $X$ and $BC$ at $Y$, then $XB_2=2$ and $YB_2=k$. Find $k^2$. [i]2017 CCA Math Bonanza Individual Round #15[/i]

In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. [b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b] In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color. ([i]Edited by Orlando Döhring[/i]) [i]Author: Radu Gologan and Dan Schwarz, Romania[/i]

Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.