This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3597

2011 Saudi Arabia IMO TST, 3
Tags: polynomial , number theory , divide , algebra
Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.

2016 Hanoi Open Mathematics Competitions, 14
Tags: polynomial , trinomial , algebra
Let $f (x) = x^2 + px + q$, where $p, q$ are integers. Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.

2013 NIMO Problems, 10
Tags: polynomial , algebra
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$. [i]Proposed by Evan Chen[/i]

2008 Greece Team Selection Test, 1
Tags: polynomial , divisibility , algebra
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$

2018 CCA Math Bonanza, TB2
Tags: polynomial , algebra
Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of \[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\] [i]2018 CCA Math Bonanza Tiebreaker Round #2[/i]

2004 India IMO Training Camp, 2
Tags: polynomial , algebra
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$

2023 IFYM, Sozopol, 8
Tags: polynomial , algebra
Do there exist a natural number $n$ and real numbers $a_0, a_1, \dots, a_n$, each equal to $1$ or $-1$, such that the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ is divisible by the polynomial $x^{2023} - 2x^{2022} + c$, where: \\ (a) $c = 1$ \\ (b) $c = -1$? [i] (For polynomials $P(x)$ and $Q(x)$ with real coefficients, we say that $P(x)$ is divisible by $Q(x)$ if there exists a polynomial $R(x)$ with real coefficients such that $P(x) = Q(x)R(x)$.)[/i]

1999 Slovenia National Olympiad, Problem 2
Tags: algebra , polynomial
Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$.

1991 India National Olympiad, 7
Tags: quadratic , complex number , rational root theorem , system of equations , polynomial , algebra
Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

2014 USAMTS Problems, 5:
Tags: invariant , algebra , polynomial , algorithm , rectangle , geometry
A finite set $S$ of unit squares is chosen out of a large grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.

2020 OMpD, 3
Tags: prime number , number theory , algebra , polynomial
Determine all integers $n$ such that both of the numbers: $$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$ are both prime numbers.

1978 AMC 12/AHSME, 13
Tags: algebra , vieta , polynomial
If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals $\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

1971 All Soviet Union Mathematical Olympiad, 149
Tags: trinomial , algebra , polynomial
Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$ then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.

1988 Vietnam National Olympiad, 2
Tags: polynomial , algebra
Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.

2023 Iran MO (2nd Round), P5
Tags: algebra , polynomial
5. We call $(P_n)_{n\in \mathbb{N}}$ an arithmetic sequence with common difference $Q(x)$ if $\forall n: P_{n+1} = P_n + Q$ $\newline$ We have an arithmetic sequence with a common difference $Q(x)$ and the first term $P(x)$ such that $P,Q$ are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that: $\newline$ a) $P(x)$ is divisible by $Q(x)$ $\newline$ b) $\text{deg}(\frac{P(x)}{Q(x)}) = 1$

2006 China Team Selection Test, 1
Tags: induction , pigeonhole principle , modular arithmetic , polynomial , algebra
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

2014 Contests, 901
Tags: calculus computations , integration , algebra , calculus , polynomial
Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

2015 Brazil National Olympiad, 5
Tags: algebra , polynomial
Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$, a polynomial $g(x)$, with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$?

2010 Indonesia TST, 1
Tags: number theory , algebra , polynomial
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

STEMS 2021 Math Cat A, Q1
Tags: algebra , polynomial
Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$. Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$. If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$, $n>0$ and $\gcd(m,n)=1$. Then find $m+n$.

2010 Postal Coaching, 1
Tags: polynomial , algebra
A polynomial $P (x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 <x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} \] Prove that the maximum value of $|P (x)|$ on the interval $[x_1 , x_n ]$ is attained in the interval $[x_{n-1} , x_n ]$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.3
Tags: polynomial , algebra
For each non-negative $a$, consider the equation $$x^3 + ax - a^3 - 29 = 0.$$ Let $x_o$ be the positive root of this equation. Prove that for all $a > 0$ such a root exists. What is the smallest value of $x_o$?

1967 AMC 12/AHSME, 16
Tags: rational root theorem , algebra , polynomial
Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is $\textbf{(A)}\ 43\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 47$

V Soros Olympiad 1998 - 99 (Russia), 11.7
Tags: polynomial , algebra
For what is the smallest natural number $n$ there is a polynomial $P(x)$ with integer coefficients, having $m$ different integer roots, and at the same time the equation $P(x) = n$ has at least one integer solution if: a) $m = 5$, b) $ m = 6$?

2023 Germany Team Selection Test, 3
Tags: number theory , prime number , polynomial , algebra
Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]