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

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Found problems: 3597

2021 Harvard-MIT Mathematics Tournament., 3
Tags: algebra , polynomial , number theory
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

1990 Turkey Team Selection Test, 2
Tags: algebra , polynomial , calculus , derivative , inequalities
For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

1978 AMC 12/AHSME, 20
Tags: polynomial
If $a,b,c$ are non-zero real numbers such that \[\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},\] and \[x=\frac{(a+b)(b+c)(c+a)}{abc},\] and $x<0$, then $x$ equals $\textbf{(A) }-1\qquad\textbf{(B) }-2\qquad\textbf{(C) }-4\qquad\textbf{(D) }-6\qquad \textbf{(E) }-8$

2003 Moldova National Olympiad, 12.5
Tags: polynomial , algebra
Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where $n\in{N^*}$. Find the remainder of the division of polynomial $P(x^{2n+1})$ by $P(x)$.

2012 NIMO Summer Contest, 9
Tags: quadratic , algebra , polynomial
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

2001 China Team Selection Test, 1
Tags: algebra , polynomial
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

2023 AMC 12/AHSME, 6
Tags: polynomial
When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive? $\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

1978 IMO Longlists, 47
Tags: polynomial , induction , algebra
Given the expression \[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\] prove: $(a) P_n(x)$ satisfies the identity \[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\] $(b) P_n(x)$ is a polynomial in $x$ of degree $n.$

2008 Putnam, B4
Tags: algebra , polynomial , modular arithmetic , function , group theory , binomial theorem
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$

2010 Contests, 2
Tags: inequalities , polynomial , algebra
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2012 India Regional Mathematical Olympiad, 6
Tags: induction , polynomial , number theory , algebra
Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

2007 Princeton University Math Competition, 2
Tags: algebra , polynomial , modular arithmetic
How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?

2020 Indonesia MO, 7
Tags: algebra , polynomial , floor function , functional equation
Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.

2010 Contests, 1
Tags: polynomial , algebra
Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

2024 Canadian Open Math Challenge, C4
Tags: algebra , polynomial
Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$. c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.

2008 Postal Coaching, 4
Tags: algebra , polynomial , root
Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.

2025 India STEMS Category A, 6
Tags: polynomial , algebra
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

2008 Iran MO (3rd Round), 5
Tags: polynomial , algebra
Prove that the following polynomial is irreducible in $ \mathbb Z[x,y]$: \[ x^{200}y^5\plus{}x^{51}y^{100}\plus{}x^{106}\minus{}4x^{100}y^5\plus{}x^{100}\minus{}2y^{100}\minus{}2x^6\plus{}4y^5\minus{}2\]

IMSC 2024, 5
Tags: algebra , polynomial , imsc
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying $$ f(xy)=P(f(x), f(y)) $$ for all $x, y\in\mathbb{R}_{>0}$.\\ [i]Proposed by Navid Safaei, Iran[/i]

1999 Ukraine Team Selection Test, 6
Tags: number theory , irreducible , polynomial
Show that for any $n \in N$ the polynomial $f(x) = (x^2 +x)^{2^n}+1$ is irreducible over $Z[x]$.

2023 Romanian Master of Mathematics Shortlist, A2
Tags: interpolation , algebra , polynomial
Fix an integer $n \geq 2$ and let $a_1, \ldots, a_n$ be integers, where $a_1 = 1$. Let $$ f(x) = \sum_{m=1}^n a_mm^x. $$ Suppose that $f(x) = 0$ for some $K$ consecutive positive integer values of $x$. In terms of $n$, determine the maximum possible value of $K$.

1983 AMC 12/AHSME, 20
Tags: trigonometry , algebra , polynomial
If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\text{(A)} \ pq \qquad \text{(B)} \ \frac{1}{pq} \qquad \text{(C)} \ \frac{p}{q^2} \qquad \text{(D)} \ \frac{q}{p^2} \qquad \text{(E)} \ \frac{p}{q}$

2019 Romania Team Selection Test, 2
Tags: algebra , polynomial , greatest common divisor , modular arithmetic , number theory
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2017 India IMO Training Camp, 1
Tags: algebra , polynomial
Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

2018 India National Olympiad, 4
Tags: polynomial , algebra , divisibility , roots of a polynomial
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.