Consider the string of length $6$ composed of three characters $a, b, c$. For each string, if two $a$s are next to each other, or two $b$s are next to each other, then replace $aa$ by $b$, and replace $bb$ by $a$. Also, if $a$ and $b$ are next to each other, or two $c$s are next to each other, remove all two of them (i.e. delete $ab, ba, cc$). Determine the number of strings that can be reduced to $c$, the string of length $1$, by the reducing processes mentioned above.

Suppose you have $15$ circles of radius $1$. Compute the side length of the smallest equilateral triangle that could possibly contain all the circles, if you are free to arrange them in any shape, provided they don’t overlap.

Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.

In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$. $\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$

Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$ [hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]

Let $ABC$ be a triangle inscribed in a circle $\omega$ and $\ell$ be the tangent to $\omega$ at $A$. The line through $B$ parallel to $AC$ meets $\ell$ at $P$, and the line through $C$ parallel to $AB$ meets $\ell$ at $Q$. The circumcircles of $ABP$ and $ACQ$ meet at $S\neq A$. Show that $AS$ bisects $BC$. [i]Proposed by Andrew Gu.[/i]

Find all the triplets of real numbers $(x , y , z)$ such that : $y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$

For each positive integer \( n \), list in increasing order all irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ 0 = \frac{p_0}{q_0} < \frac{1}{n} = \frac{p_1}{q_1} < \cdots < \frac{1}{1} = \frac{p_{M(n)}}{q_{M(n)}}. \] Let \( k \) be a positive integer. We define, for each \( n \) such that \( M(n) \geq k - 1 \), \[ f_k(n) = \min \left\{ \sum_{s=0}^{k-1} q_{j+s} : 0 \leq j \leq M(n) - k + 1 \right\}. \] Determine, in function of \( k \), \[ \lim_{n \to \infty} \frac{f_k(n)}{n}. \] For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1 \) and \( q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \). In this case, we have \( f_1(4) = 1, f_2(4) = 5, f_3(4) = 8, f_4(4) = 10, f_5(4) = 13, f_6(4) = 17 \), and \( f_7(4) = 18 \).

Let $n \geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right. [i]Proposed by Milan Haiman[/i]

Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m),$ meaning that \[ k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m, \] where each $f_i$ is an integer, $0\le f_i\le i,$ and $0<f_m.$ Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\cdots+1968!-1984!+2000!,$ find the value of $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j.$

Let $P$ be a point in parallelogram $ABCD$ such that $$PA \cdot PC + PB \cdot PD = AB \cdot BC.$$ Prove that the reflections of $P$ over lines $AB$, $BC$, $CD$, and $DA$ are concyclic.

Let $ABCD$ be a rhombus such that $m(\widehat{ABC}) = 40^\circ$. Let $E$ be the midpoint of $[BC]$ and $F$ be the foot of the perpendicular from $A$ to $DE$. What is $m(\widehat{DFC})$? $ \textbf{a)}\ 100^\circ \qquad\textbf{b)}\ 110^\circ \qquad\textbf{c)}\ 115^\circ \qquad\textbf{d)}\ 120^\circ \qquad\textbf{e)}\ 135^\circ $

We have that $(a+b)^3=216$, where $a$ and $b$ are positive integers such that $a>b$. What are the possible values of $a^2-b^2$?

Let $n$ be a positive integer. Let $x_1, x_2, \dots, x_n > 1$ be real numbers whose product is $n+1$. Prove that \[\left(\frac{1}{1^2(x_1-1)}+1\right)\left(\frac{1}{2^2(x_2-1)}+1\right)\cdots\left(\frac{1}{n^2(x_n-1)}+1\right)\geq n+1\] and find for which values equality holds.

Given $a+b+c=5$ and $ 1 \le a, b, c \le 2 $, what is the minimum possible value of $\frac{1}{a+b}+\frac{1}{b+c}$?

In acute triangle $ABC$, $D\in BC,E\in CA,F\in AB$. Prove that the necessary and sufficient condition of $AD,BE,CF$ are heights of $\triangle ABC$ is that $S=\frac{R}{2}(EF+FD+DE)$. Note: $S$ is the area of $\triangle ABC$, $R$ is the circumradius of $\triangle ABC$.

Let $p \geq 5$ be a prime number. Prove that there exist at least 2 distinct primes $q_1, q_2$ satisfying $1 < q_i < p - 1$ and $q_i^{p-1} \not\equiv 1 \mbox{ (mod }p^2)$, for $i = 1, 2$.

Let be a real number $ r $ and two functions $ f:[r,\infty )\longrightarrow\mathbb{R} , F_1:(r,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties. $ \text{(i)} f $ has Darboux's intermediate value property. $ \text{(ii)} F_1$ is differentiable and $ F'_1=f\bigg|_{(r,\infty )} $ [b]1)[/b] Provide an example of what $ f,F_1 $ could be if $ f $ hasn't a lateral limit at $ r, $ and $ F_1 $ has lateral limit at $ r. $ Moreover, if $ f $ has lateral limit at $ r, $ show that [b]2)[/b] $ F_1 $ has a finite lateral limit at $ r. $ [b]3)[/b] the function $ F:[r,\infty )\longrightarrow\mathbb{R} $ defined as $$ F(x)=\left\{ \begin{matrix} F_1(x) ,& \quad x\in (r,\infty ) \\ \lim_{\stackrel{x\to r}{x>r}} F_1(x), & \quad x=r \end{matrix} \right. $$ is a primitive of $ f. $

For a rectangle $R$ with integral side lengths, denote by $D(a, b)$ the number of ways of covering $R$ by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of $R$. Determine the perimeter $P$ of the rectangle $R$ for which $\frac{D(a,b)}{a+b}$ is maximal.

Given a positive real $t$, a set $S$ of nonnegative reals is called $t$-good if for any two distinct elements $a,b$ in $S$, $\frac{a+b}2\ge\sqrt{ab}+t$. For all positive reals $N$, find the maximum number of elements a $t$-good set can have, if all elements are at most $N$. [i](Proposed by Ho Janson)[/i]

You and five friends need to raise $ \$1500$ in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? $ \textbf{(A)}\ 250\qquad \textbf{(B)}\ 300\qquad \textbf{(C)}\ 1500\qquad \textbf{(D)}\ 7500\qquad \textbf{(E)}\ 9000$

Arne and Bertil play a game on an $11 \times 11$ grid. Arne starts. He has a game piece that is placed on the center od the grid at the beginning of the game. At each move he moves the piece one step horizontally or vertically. Bertil places a wall along each move any of an optional four squares. Arne is not allowed to move his piece through a wall. Arne wins if he manages to move the pice out of the board, while Bertil wins if he manages to prevent Arne from doing that. Who wins if from the beginning there are no walls on the game board and both players play optimally?

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]