Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer. [i]Radu Gologan[/i]

The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition \[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\] for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.

$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.

For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$? [i]2018 CCA Math Bonanza Individual Round #12[/i]

Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties. 1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$ 2. $w$ contains an even number of $a$'s 3. $w$ contains an even number of $b$'s. For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that $$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.

Real numbers $ x_{1},x_{2},..,x_{n}$ are positive. Prove the inequality: $ \frac{x_{1}}{x_{2}\plus{}x_{3}}\plus{}\frac{x_{2}}{x_{3}\plus{}x_{4}}\plus{}...\plus{} \frac{x_{n\minus{}1}}{x_{n}\plus{}x_{1}}\plus{}\frac{x_{n}}{x_{1}\plus{}x_{2}}>(\sqrt{2}\minus{}1)n$

For natural number $t$, the repeating base-$t$ representation of the (base-ten) rational number $\frac{7}{51}$ is $0.\overline{23}_t=0.232323..._t$. What is $t$ ?

Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true? [asy]/* AMC8 2002 #16 Problem */ draw((0,0)--(4,0)--(4,3)--cycle); draw((4,3)--(-4,4)--(0,0)); draw((-0.15,0.1)--(0,0.25)--(.15,0.1)); draw((0,0)--(4,-4)--(4,0)); draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2)); draw((4,0)--(7,3)--(4,3)); draw((4,2.8)--(4.2,2.8)--(4.2,3)); label(scale(0.8)*"$Z$", (0, 3), S); label(scale(0.8)*"$Y$", (3,-2)); label(scale(0.8)*"$X$", (5.5, 2.5)); label(scale(0.8)*"$W$", (2.6,1)); label(scale(0.65)*"5", (2,2)); label(scale(0.65)*"4", (2.3,-0.4)); label(scale(0.65)*"3", (4.3,1.5));[/asy] $ \textbf{(A)}\ X\plus{}Z\equal{}W\plus{}Y \qquad \textbf{(B)}\ W\plus{}X\equal{}Z \qquad\textbf{(C)}\ 3X\plus{}4Y\equal{}5Z \qquad $ $\textbf{(D)}\ X\plus{}W\equal{}\frac{1}{2}(Y\plus{}Z) \qquad\textbf{(E)}\ X\plus{}Y\equal{}Z$

What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of above} $

Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds. a) Prove that: \[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)} \] b) Prove that \[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1) \] c) Prove the statement of problem. d) Prove Van der Waerden's Theorem.

A positive integer is called [i]shiny[/i] if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is [i]shiny[/i], because $2012 = 2005 + 7$, and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are [b]not[/b] [i]shiny[/i] (the dark integers).

Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, ... , x_7) = f(x_1) + f(x_2) +... + f(x_7)$. If $x_1, x_2,..., x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, ... , x_7)$ does $g(x_1, x_2, ... , x_7)$ obtain its maximal value.

If $ x$,$ y$, and $ z$ are real numbers such that \[(x \minus{} 3)^2 \plus{} (y \minus{} 4)^2 \plus{} (z \minus{} 5)^2 \equal{} 0,\] then $ x \plus{} y \plus{} z \equal{}$ $ \textbf{(A)}\ \minus{}12\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 50$

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? [i]Nikolai Beluhov[/i]

It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.

From $2008 \times 2008$ square we remove one corner cell $1 \times 1$. Is number of ways to divide this figure to corners from $3$ cells odd or even ?

[b](a)[/b] Determine if exist an integer $n$ such that $n^2 -k$ has exactly $10$ positive divisors for each $k = 1, 2, 3.$ [b](b) [/b]Show that the number of positive divisors of $n^2 -4$ is not $10$ for any integer $n.$

$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$. [i]Proposed by Bradley Guo[/i]

A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.

Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$. (Proposed by Gerhard Woeginger, Austria)

Consider the lattice of all algebraically closed subfields of the complex field $ \mathbb{C}$ whose transcendency degree (over $ \mathbb{Q}$) is finite. Prove that this lattice is not modular. [i]L. Babai[/i]

Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.