Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression: \[p(1)+p(2)+p(3)+...+p(999).\]

Let $ABC$ be a triangle with incenter $I$. The line through $I$, perpendicular to $AI$, intersects the circumcircle of $ABC$ at points $P$ and $Q$. It turns out there exists a point $T$ on the side $BC$ such that $AB + BT = AC + CT$ and $AT^2 = AB \cdot AC$. Determine all possible values of the ratio $IP/IQ$.

Prove:If $ a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n}$ is a perfect number, then $ 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4$ ; if moreover, $ a$ is odd, then the upper bound $ 4$ may be reduced to $ 2\sqrt[3]{2}$.

7. In triangle $ABC$, let $N$ and $M$ be the midpoints of $AB$ and $AC$, respectively. Point $P$ is chosen on the arc $BC$ not containing $A$ of the circumcircle of $ABC$ such that $BNMP$ is cyclic. Given $BC=28$, $AC=30$ and $AB = 26$, the value of $AP$ may be expressed as $m/\sqrt{n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Compute $m+n$. [i]Proposed by winnertakeover[/i]

Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?

for any positive integer $n$ greater than $1$, show that \[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]

Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows: $+$ $0$ $a$ $b$ $c$ $0$ $0$ $a$ $b$ $c$ $a$ $a$ $0$ $c$ $b$ $b$ $b$ $c$ $0$ $a$ $c$ $c$ $b$ $a$ $0$ a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$. b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.

We have a deck of $2n$ cards. Each shuffling changes the order from $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ to $a_1,b_1,a_2,b_2,\ldots,a_n,b_n$. Determine all even numbers $2n$ such that after shuffling the deck $8$ times the original order is restored.

Heesu plays a game where he starts with $1$ piece of candy. Every turn, he flips a fair coin. On heads, he gains another piece of candy, unless he already has $5$ pieces of candy, in which case he loses $4$ pieces of candy and goes back to having $1$ piece of candy. On tails, the game ends. What is the expected number of pieces of candy that Heesu will have when the game ends?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$

Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.

Let $n$ be a positive integer. Show that there exists an integer $m$ such that \[ 2018m^2+20182017m+2017 \] is divisible by $2^n$.

Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon $ABCDEF$. Then Dylan chooses a point $P$ within the circle of radius $2$ centered at the origin. Let $M$ be the maximum possible value of $|PA| \cdot |PB| \cdot |PC| \cdot |PD| \cdot |PE| \cdot |PF|$, and let $N$ be the number of possible points $P$ for which this maximal value is obtained. Find $M + N^2$.

Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$

There are $2^{2018}$ positions on a circle numbered from $1$ to $2^{2018}$ in a clockwise manner. Initially, two white marbles are placed at positions $2018$ and $2019$. Before the game starts, Ping chooses to place either a black marble or a white marble at each remaining position. At the start of the game, Ping is given an integer $n$ ($0\leq n\leq 2018$) and two marbles, one black and one white. He will then move around the circle, starting at position $2n$ and moving clockwise by $2n$ positions at a time. At the starting position and each position he reaches, Ping must switch the marble at that position with a marble of the other color he carries. If he cannot do so at any position, he loses the game. Is there a way to place the $2^{2018}-2$ remaining marbles so that Ping will never lose the game regardless of the number $n$ and the number of rounds he moves around the circle?

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

The NIMO problem writers have invented a new chess piece called the [i]Oriented Knight[/i]. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16\times 16$ chess board? [i] Proposed by Tony Kim and David Altizio [/i]

Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.

$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Let $ABC$ be a scalene triangle with $\Omega_A, \Omega_B,\Omega_C$ its excircles. $T_A$ is the intersection point of the external tangent (different of $AB$) of $\Omega_A,\Omega_B$ with the external tangent (different of $AC$) of $\Omega_A, \Omega_C$. Define $T_B, T_C$ in a similar way. If $I_A, I_B, I_C$ are the excenters of $ABC$, prove that the circumcircles of $AI_AT_A, BI_BT_B, CI_CT_C$ concur in exactly two points.

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties: [list] [*] $f(1,1)=0$. [*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1; [*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$. [/list] Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$

A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.