Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.

If Scott rolls four fair six-sided dice, what is the probability that he rolls more 2's than 1's? $\text{(A) }\frac{8}{27}\qquad\text{(B) }\frac{25}{81}\qquad\text{(C) }\frac{103}{324}\qquad\text{(D) }\frac{421}{1296}\qquad\text{(E) }\frac{65}{162}$

How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.

Beter Pai wants to tell you his fastest $40$-line clear time in Tetris, but since he does not want Qep to realize she is better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements, given that the correct time is t seconds: $\bullet$ $t < 100$. $\bullet$ $t$ is prime. $\bullet$ $t -1$ has 5 proper factors. $\bullet$ all prime factors of $t +1$ are single digits. $\bullet$ $t -2$ is a multiple of $3$. $\bullet$ $t +2$ has $2$ factors. Find t.

The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.

Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$ Here $d(n)$ is the number of positive divisors of a positive integer

Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{max}$ and $P_{min}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{max}$ and $P_{min}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$. [i]Proposed by [b]FedeX333X [/b][/i]

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 $

There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5: 4: 3,$ you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?

Find all positive integers n such that the product $1! \cdot 2! \cdot \cdot \cdot \cdot n!$ is a perfect square

Eight men had participated in the chess tournament. (Each meets each, draws are allowed, giving $1/2$ of point, winner gets $1$.) Everyone has different number of points. The second one has got as many points as the four weakest participants together. What was the result of the play between the third prizer and the chess-player that have occupied the seventh place?

Kevin is in kindergarten, so his teacher puts a $100 \times 200$ addition table on the board during class. The teacher first randomly generates distinct positive integers $a_1, a_2, \dots, a_{100}$ in the range $[1, 2016]$ corresponding to the rows, and then she randomly generates distinct positive integers $b_1, b_2, \dots, b_{200}$ in the range $[1, 2016]$ corresponding to the columns. She then fills in the addition table by writing the number $a_i+b_j$ in the square $(i, j)$ for each $1\le i\le 100$, $1\le j\le 200$. During recess, Kevin takes the addition table and draws it on the playground using chalk. Now he can play hopscotch on it! He wants to hop from $(1, 1)$ to $(100, 200)$. At each step, he can jump in one of $8$ directions to a new square bordering the square he stands on a side or at a corner. Let $M$ be the minimum possible sum of the numbers on the squares he jumps on during his path to $(100, 200)$ (including both the starting and ending squares). The expected value of $M$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q.$ [i]Proposed by Yang Liu[/i]

It is about deciphering the code $C_1C_2C_3C_4$ in which each letter is one of the colors: white $(B)$, blue $(A)$, red $(R)$, green $(V)$, black $(N)$ and brown $(C)$ with allowed repetitions. Four were made attempts to decipher it. $NAVB$ and $ACRC$ have two color hits, but in wrong places. $RBAC$ and $VRBA$ have one color match in the correct place, and two other color matches, in places incorrect. Determine all combinations compatible with the information.