Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

The [i]minimal polynomial[/i] of a complex number $r$ is the unique polynomial with rational coefficients of minimal degree with leading coefficient $1$ that has $r$ as a root. If $f$ is the minimal polynomial of $\cos\frac\pi7$, what is $f(-1)$?

[b]p1.[/b] Compute $1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55$. [b]p2.[/b] Given that $a$, $b$, and $c$ are positive integers such that $a+b = 9$ and $bc = 30$, find the minimum possible value of $a + c$. [b]p3.[/b] Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ . [b]p4.[/b] Let $N$ be the product of the positive integer divisors of $8!$, including itself. The largest integer power of $2$ that divides $N$ is $2^k$. Compute $k$. [b]p5.[/b] Let $A=(-20, 22)$, $B = (k, 0)$, and $C = (202, 2)$ be points on the coordinate plane. Given that $\angle ABC = 90^o$, find the sum of all possible values of $k$. [b]p6.[/b] Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ satisfy both $a+b-c = 12$ and $a^2+b^2-c^2 = 24$? [b]p8.[/b] For how many three-digit base-$7$ numbers $\overline{ABC}_7$ does $\overline{ABC}_7$ divide $\overline{ABC}_{10}$? (Note: $\overline{ABC}_D$ refers to the number whose digits in base $D$ are, from left to right, $A$, $B$, and $C$; for example, $\overline{123}_4$ equals $27$ in base ten). [b]p9.[/b] Natasha is sitting on one of the $35$ squares of a $5$-by-$7$ grid of squares. Wanda wants to walk through every square on the board exactly once except the one Natasha is on, starting and ending on any $2$ squares she chooses, such that from any square she can only go to an adjacent square (two squares are adjacent if they share an edge). How many squares can Natasha choose to sit on such that Wanda cannot go on her walk? [b]p10.[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Point $P$ lies inside $ABC$ and points $D,E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, so that $PD \perp BC$, $PE \perp CA$, and $PF \perp AB$. Given that $PD$, $PE$, and $PF$ are all integers, find the sum of all possible distinct values of $PD \cdot PE \cdot PF$. [b]p11.[/b] A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$. [b]p12.[/b] Let $P(x) = x^2+px+q$ be a quadratic polynomial with positive integer coefficients. Compute the least possible value of p such that 220 divides p and the equation $P(x^3) = P(x)$ has at least four distinct integer solutions. [b]p13.[/b] Everyone at a math club is either a truth-teller, a liar, or a piggybacker. A truth-teller always tells the truth, a liar always lies, and a piggybacker will answer in the style of the previous person who spoke (i.e., if the person before told the truth, they will tell the truth, and if the person before lied, then they will lie). If a piggybacker is the first one to talk, they will randomly either tell the truth or lie. Four seniors in the math club were interviewed and here was their conversation: Neil: There are two liars among us. Lucy: Neil is a piggybacker. Kevin: Excluding me, there are more truth-tellers than liars here. Neil: Actually, there are more liars than truth-tellers if we exclude Kevin. Jacob: One plus one equals three. Define the base-$4$ number $M = \overline{NLKJ}_4$, where each digit is $1$ for a truth-teller, $2$ for a piggybacker, and $3$ for a liar ($N$ corresponds to Neil, $L$ to Lucy, $K$ corresponds to Kevin, and $J$ corresponds to Jacob). What is the sum of all possible values of $M$, expressed in base $10$? [b]p14.[/b] An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells, then that cell becomes living, and any living cell remains living. Given that every cell in the grid eventually evolves to be living, what is the minimum possible number of living cells in the initial grid? [b]p15.[/b] In triangle $ABC$, $AB = 7$, $BC = 11$, and $CA = 13$. Let $\Gamma$ be the circumcircle of $ABC$ and let $M$, $N$, and $P$ be the midpoints of minor arcs $BC$ , $CA$, and $AB$ of $\Gamma$, respectively. Given that $K$ denotes the area of $ABC$ and $L$ denotes the area of the intersection of $ABC$ and $MNP$, the ratio $L/K$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

Real numbers $a_1,a_2,...,a_{16}$ satisfy the conditions $\sum_{i=1}^{16}a_i = 100$ and $\sum_{i=1}^{16}a_i^2 = 1000$ . What is the greatest possible value of $a_16$?

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$ a) is finite b) does not exceed $n$.

Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .

Determine all numbers $x, y$ and $z$ satisfying the system of equations $$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$

The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality: $3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.

Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$

Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.

Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

Solve the system of equations \[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]

Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

How many real numbers $a$ are there for which both solutions to the equation \[x^2 + (a - 2024)x + a = 0\] are integers? \[\mathrm a. ~15\qquad \mathrm b. ~16 \qquad \mathrm c. ~18 \qquad\mathrm d. ~20\qquad\mathrm e. ~24\qquad\]

For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational. [list](a) Prove the claim for $r \ge \frac43$ and $r \le 0$. (b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold. [/list]