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.

AND:
OR:
NO:

Found problems: 3597

1991 Baltic Way, 8
Tags: calculus , derivative , algebra , polynomial
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation \[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\] \[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\] \[+(x - b)(x - c)(x - d)(x - e) = 0\] has four distinct real solutions.

2005 AMC 12/AHSME, 24
Tags: function , quadratic , algebra , polynomial
Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$? $ \textbf{(A)}\ 19\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 32$

1970 IMO Longlists, 47
Tags: derivative , algebra , calculus , polynomial
Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

2013 Purple Comet Problems, 27
Tags: polynomial , algebra
Suppose $a,b$ and $c$ are real numbers that satisfy $a+b+c=5$ and $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac15$. Find the greatest possible value of $a^3+b^3+c^3$.

2004 All-Russian Olympiad Regional Round, 11.3
Tags: polynomial , algebra
Let the polynomial $P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ has at least one real root and $a_0 \ne 0$. Prove that, consequently crossing out the monomials in the notation $P(x)$ in some order, we can obtain the number $a_0$ from it so that each intermediate polynomial also has at least one real root.

KoMaL A Problems 2023/2024, A. 867
Tags: algebra , polynomial
Let $p(x)$ be a monic integer polynomial of degree $n$ that has $n$ real roots, $\alpha_1,\alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$. Prove that \[\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.\] [i]Submitted by Dávid Matolcsi, Berkeley[/i]

1987 India National Olympiad, 6
Tags: polynomial , algebra , quadratic , greatest common divisor , rational root theorem
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.

2016 Saudi Arabia BMO TST, 3
Tags: algebra , polynomial , integer polynomial
Does there exist a polynomial $P(x)$ with integral coefficients such that a) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ? b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?

1997 Romania National Olympiad, 1
Tags: algebra , polynomial
Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$ Prove that: a) the polynomial has no integer root; β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.

2022 Greece Junior Math Olympiad, 1
Tags: polynomial , inequalities , algebra
(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients. (b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$ find the maximum possible value of $a$.

1940 Putnam, A6
Tags: polynomial , algebra
Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$. Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.

1988 IMO Longlists, 38
Tags: algebra , polynomial
[b]i.)[/b] The polynomial $x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}$ is not divisible by $x^2 + x + 1.$ Find the value of $k.$ [b]ii.)[/b] If $p,q$ and $r$ are distinct roots of $x^3 - x^2 + x - 2 = 0$ the find the value of $p^3 + q^3 + r^3.$ [b]iii.)[/b] If $r$ is the remainder when each of the numbers 1059, 1417 and 2312 is divided by $d,$ where $d$ is an integer greater than one, then find the value of $d-r.$ [b]iv.)[/b] What is the smallest positive odd integer $n$ such that the product of \[ 2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}} \] is greater than 1000?

2005 Bulgaria National Olympiad, 6
Tags: polynomial , algebra , symmetry , number theory , modular arithmetic
Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.

KoMaL A Problems 2022/2023, A. 851
Tags: algebra , polynomial , counting , combinatorial geometry , combinatorics
Let $k$, $\ell $ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^\circ$ and let its sidelengths be $AB=k$, $BC=\ell$ and $CD=m$. Let $f(k,\ell,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^\circ$. Prove that by fixing $\ell$ and $m$, there exists polynomial $g_{\ell,m}$ such that $f(k,\ell,m)=g_{\ell,m}(k)$ for every positive integer $k$, and find the degree of $g_{\ell,m}$ in terms of $\ell$ and $m$. [i]Submitted by Zoltán Gyenes, Budapest[/i]

2021 Estonia Team Selection Test, 2
Tags: polynomial , algebra
Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

1970 Putnam, B2
Tags: polynomial , algebra
The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

1985 ITAMO, 13
Tags: algorithm , polynomial , number theory , greatest common divisor , euclidean algorithm , algebra
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

1998 Bulgaria National Olympiad, 2
Tags: polynomial , algebra
The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.

1994 Denmark MO - Mohr Contest, 3
Tags: polynomial , algebra
The third-degree polynomial $P(x)=x^3+2x^2-3x-5$ has the three roots $a$, $b$ and $c$. State a third degree polynomial with roots $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$.

2015 Germany Team Selection Test, 1
Tags: polynomial , algebra , root , real number , integer
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

2019 India PRMO, 23
Tags: algebra , polynomial
Let $t$ be the area of a regular pentagon with each side equal to $1$. Let $P(x)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $\gcd$ of all the coefficients equal to $1$. If $M$ is the sum of the absolute values of the coefficients of $P(x)$, What is the integer closest to $\sqrt{M}$ ? ($\sin 18^{\circ}=(\sqrt{5}-1)/2$)

Kvant 2023, M2754
Tags: polynomial , algebra
Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root. Proposed by N. Agakhanov

2012 ELMO Shortlist, 3
Tags: algebra , polynomial , function
Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

2019 Thailand TSTST, 3
Tags: polynomial , combinatorics , algebra , coefficient
Let $n\geq 2$ be an integer. Determine the number of terms in the polynomial $$\prod_{1\leq i< j\leq n}(x_i+x_j)$$ whose coefficients are odd integers.

2016 IFYM, Sozopol, 6
Tags: polynomial , algebra
Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.