Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.

[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$. [b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$. [b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$, $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$ [b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$. [b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$. [b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations $$x^2 + y^2 \le 2$$ $$x + y + |z| \le 3.$$ [b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$. [b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$ (Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.) [b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$. [b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $|a + b\omega + c\omega^2|$.  

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.

Maria has a $6 \times 5$ board with some shaded squares, as in the figure. She writes, in some order, the digits $1, 2, 3, 4$ and $5$ in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row. For example, if you wrote $2, 3, 4, 1, 5$ in the first row, then since $4$ is in the shaded box, the number that occupies the fourth place $(1)$ is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: $2, 3, 4, 5, 1$. Then, to complete the third row, as in the shaded box is $3$, the number located in the third place $(4)$ writes it in the last box and gets $2, 3, 5, 1, 4$. Following in the same way, he gets the board of the figure. Show a way to locate the numbers in the first row to get the numbers $2, 4, 5, 1, 3$ in the last row.

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Let $n\in {{\mathbb{N}}^{*}}$ and ${{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}$ so ${{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.$Prove that $\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}$

The roots of the equation $x^5-180x^4+Ax^3+Bx^2+Cx+D=0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that: [b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively. [b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.

Consider a table of cyclic permutations ($n\ge2$) \[ \begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix} \] Then multiply each number of the first row by that number of the $k$-th row that is in the same column. Sum all these products and denote $s_k$ the result (e.g. $s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1$). a) Find a recursive relation for $s_k$ in terms of $s_{k-1}$ and determine the explicit formula for $s_k$. b) Determine both an index $k$ and the value of $s_k$ such that the sum $s_k$ is minimal.

Given an isosceles triangle $ABC$ with base $AB$, cirumcenter $O$, incenter $S$ and $\angle C<60^\circ$. The circumcircle of $AOS$ intersects $AC$ at $D$. Prove that $SD\parallel BC$ and $AS\perp OD$.

Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that $$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$ a) Prove that there exist a prime number which divides infinitely many $a_n$ b) Prove that there exist infinitely many such prime numbers

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

A group of nine math team members like to play Survev.io. They noticed that the number of hours each of them played this week forms an arithmetic progression. The person who played the least played for $1$ hour, while the most played for $9.$ Find the total number of hours all nine group members spent playing Survev.io this week.

We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.