Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1\] holds. If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

We are given an acute triangle $ABC$. The angle bisector of $\angle BAC$ cuts $BC$ at $P$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively, so that $BC \parallel DE$. Points $K$ and $L$ lie on segments $PD$ and $PE$, respectively, so that points $A$, $D$, $E$, $K$, $L$ are concyclic. Prove that points $B$, $C$, $K$, $L$ are also concyclic. [i]Proposed by Patrik Bak, Slovakia [/i]

$16$ different natural numbers are written on the board, none of which exceeds $30$. Prove that there must be two coprime numbers among the written numbers.

[b]a)[/b] Prove that, for any point in the interior of a triangle, there are two points on the sides of this triangle such that the resultant of the vectors from the interior point those two points is the vector $ 0. $ [b]b)[/b] Prove that, for any point in the interior of a triangle, there are three points on the sides of this triangle such that the resultant of the vectors from the interior point those three points is the vector $ 0. $

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Given is a trihedral angle $ OABC $ with a vertex $ O $ and a point $ P $ in its interior. Let $ V $ be the volume of a parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Calculate the minimum volume of a tetrahedron whose three faces are contained in the faces of the trihedral angle $OABC$ and the fourth face contains the point $P$.

The fraction $ \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}}$ is equal to: $ \textbf{(A)}\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad \textbf{(B)}\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad \textbf{(C)}\ a^{\minus{}2}\plus{}b^{\minus{}2} \\ \textbf{(D)}\ a^2\plus{}b^2 \qquad \textbf{(E)}\ a^2\minus{}b^2$

Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?

A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even. [img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]

Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$

An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].

Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Let $O$ and $H$ be the circumcenter and the orthocenter respectively of triangle $ABC$. Itis known that $BH$ is the bisector of angle $ABO$. The line passing through $O$ and parallel to $AB$ meets $AC$ at $K$. Prove that $AH = AK$

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that \[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]

Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line. [i]By Aliakbar Daemi[/i]

Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$, where $\sigma(n)$ is the sum of all positive divisors of $n$.

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)? (A Andzans)

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

Prove that there exists a positive integer $n$, such that if an equilateral triangle with side lengths $n$ is split into $n^2$ triangles with side lengths 1 with lines parallel to its sides, then among the vertices of the small triangles it is possible to choose $1993n$ points so that no three of them are vertices of an equilateral triangle.

[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$. [b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines. [b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$. [b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.) [b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$. [b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$. [b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$. [b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this. [b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer. [b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings. [b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$. [b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$. [b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$. [b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$. [img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img] [b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$) The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.