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.

Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square? [asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2")); dot("$A$",A,W,linewidth(4)); dot("$B$",B,dir(0),linewidth(4)); dot("$C$",C,dir(0),linewidth(4)); dot("$D$",D,dir(20),linewidth(4)); dot("$E$",E,dir(100),linewidth(4)); dot("$F$",F,W,linewidth(4)); dot("$X$",X,dir(0),linewidth(4)); dot("$Y$",Y,N,linewidth(4)); dot("$Z$",Z,W,linewidth(4)); [/asy] $ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$ $\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3} \qquad\textbf{(E)}\ 21\sqrt{6}$

Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values ​​of the number $k$ if $n=d_2d_3 + d_2d_5+d_3d_5$.

$ \log 125$ equals: $ \textbf{(A)}\ 100 \log 1.25 \qquad \textbf{(B)}\ 5 \log 3 \qquad \textbf{(C)}\ 3 \log 25$ $ \textbf{(D)}\ 3 \minus{} 3\log 2 \qquad \textbf{(E)}\ (\log 25)(\log 5)$

Let $ABC$ be a triangle with given sides $a,b,c$. Determine the minimal possible length of the diagonal of an inscribed rectangle in this triangle. [i]Note: A rectangle is inscribed in the triangle if two of its consecutive vertices lie on one side of the triangle, while the other two vertices lie on the other two sides of the triangle. [/i]

Some $n > 2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ... , lamp $k$ with $k+1$, ... , lamp $n$ with lamp $1$. At the beginning, all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice-versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.