Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? $ \textbf{(A)}\ \frac{1}{16}\qquad \textbf{(B)}\ \frac18\qquad \textbf{(C)}\ \frac{3}{16} \qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac{5}{16}$

Find all integers $k$, such that there exists an integer sequence ${\{a_n\}}$ satisfies two conditions below (1) For all positive integers $n$,$a_{n+1}={a_n}^3+ka_n+1$ (2) $|a_n| \leq M$ holds for some real $M$

Given that \[ \sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!} \] for some positive integer $m$, find $m \pmod{71}$.

On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.

We have a $9$ by $9$ chessboard with $9$ kings (which can move to any of $8$ adjacent squares) in the bottom row. What is the minimum number of moves, if two pieces cannot occupy the same square at the same time, to move all the kings into an $X$ shape (a $5\times5$ region where there are $5$ kings along each diagonal of the $X$, as shown below)? \begin{tabular}{ c c c c c } O & & & & O \\ & O & & O & \\ & & O & & \\ & O & & O & \\ O & & & & O \\ \end{tabular} [i]Proposed by David Tang[/i]

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

Let $ A $ be a nonempty set of real numbers, and let be two functions $ f,g:A\longrightarrow A $ having the following properties: $ \text{(i)} f $ is increasing $ \text{(ii)} f-g $ is nonpositive everywhere $ \text{(iii)} f(A)\subset g(A) $ [b]a)[/b] Prove that $ f=g $ if $ A $ is the set of all nonnegative integers. [b]b)[/b] Is true that $ f=g $ if $ A $ is the set of all integers? [i]Dorel Miheț[/i]

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$. (Two queens attack each other when they have different colors. The queens of the same color don't attack each other)

A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.

Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.

A triangle has sides of length $9$, $40$, and $41$. What is its area?

Denote the set of all positive integers by $\mathbb{N}$, and the set of all ordered positive integers by $\mathbb{N}^2$. For all non-negative integers $k$, define [i]good functions of order k[/i] recursively for all non-negative integers $k$, among all functions from $\mathbb{N}^2$ to $\mathbb{N}$ as follows: (i) The functions $f(a,b)=a$ and $f(a,b)=b$ are both good functions of order $0$. (ii) If $f(a,b)$ and $g(a,b)$ are good functions of orders $p$ and $q$, respectively, then $\gcd(f(a,b),g(a,b))$ is a good function of order $p+q$, while $f(a,b)g(a,b)$ is a good function of order $p+q+1$. Prove that, if $f(a,b)$ is a good function of order $k\leq \binom{n}{3}$ for some positive integer $n\geq 3$, then there exist a positive integer $t\leq \binom{n}{2}$ and $t$ pairs of non-negative integers $(x_1,y_1),\ldots,(x_n,y_n)$ such that $$f(a,b)=\gcd(a^{x_1}b^{y_1},\ldots,a^{x_t}b^{y_t})$$ holds for all positive integers $a$ and $b$. [i]Proposed by usjl[/i]

Determine the minimum value of $f (x)$ where f (x) = (3 sin x - 4 cos x - 10)(3 sin x + 4 cos x - 10).

Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 19 $

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

Let $m$ be a positive integer and $a$ and $b$ be distinct positive integers strictly greater than $m^2$ and strictly less than $m^2+m$. Find all integers $d$ such that $m^2 < d < m^2+m$ and $d$ divides $ab$.

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

Three players $A, B$ and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, $....$, $A$ starts the game and the one who takes the last stone loses. Players $A$ and $C$ They form a team against $B$, they agree on a strategy joint. $B$ can take $1, 2, 3, 4$ or $5$ stones on each move, while that $A$ and $C$ can each draw $1, 2$ or $3$ stones in each turn. Determine for which values of $N$ have winning strategies $A$ and $C$ , and for what values the winning strategy is $B$'s.

Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]

Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares.

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]