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

1969 IMO Longlists, 14
Tags: quadratic , algebra , inequality , root , polynomial
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

2017 IMO Shortlist, A1
Tags: algebra , polynomial , ineqstd
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

2022 Kyiv City MO Round 2, Problem 2
Tags: algebra , polynomial
Initially memory of computer contained a single polynomial $x^2-1$. Every minute computer chooses any polynomial $f(x)$ from its memory and writes $f(x^2-1)$ and $f(x)^2-1$ to it, or chooses any two distinct polynomials $g(x), h(x)$ from its memory and writes polynomial $\frac{g(x) + h(x)}{2}$ to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains $P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1$? [i](Proposed by Arsenii Nikolaiev)[/i]

2005 Putnam, B5
Tags: algebra , polynomial , calculus , derivative , system of equations
Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

1998 Baltic Way, 8
Tags: polynomial , algebra
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that \[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \] for every real number $x$ and every positive integer $n$.

2011 Bogdan Stan, 1
Tags: group theory , abstract algebra , algebra , polynomial , linear algebra , matrix
Consider the multiplicative group $ \left\{ \left.A_k:=\left(\begin{matrix} 2^k& 2^k\\2^k& 2^k\end{matrix}\right)\right| k\in\mathbb{Z} \right\} . $ [b]a)[/b] Prove that $A_xA_y=A_{x+y+1} , $ for all integers $ x,y. $ [b]b)[/b] Show that, for all integers $ t, $ the multiplicative group $ \left\{ A_{jt-1}|j\in\mathbb{Z} \right\} $ is a subgroup of $ G. $ [b]c)[/b] Determine the linear integer polynomials $ P $ for which it exists an isomorphism $ \left( G,\cdot \right)\stackrel{\eta}{\cong}\left( \mathbb{Z} ,+ \right) $ such that $ \eta\left( A_k \right) =P(k). $

2014 IFYM, Sozopol, 4
Tags: polynomial , algebra
Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and $(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.

2008 239 Open Mathematical Olympiad, 6
Tags: polynomial , algebra , function
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies $$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies $$P(x,y) = q(x+y)-q(x)-q(y)$$

2010 Germany Team Selection Test, 1
Tags: polynomial , algebra
Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

2004 Baltic Way, 8
Tags: algebra , polynomial , inequalities , function , number theory , prime number , logarithm
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2004 Romania Team Selection Test, 6
Tags: number theory , invariant , symmetry , quadratic , algebra , polynomial , vieta
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

1996 IMO Shortlist, 4
Tags: inequalities , polynomial , function , calculus , algebra
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

1999 All-Russian Olympiad Regional Round, 11.8
Tags: algebra , polynomial
For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.

2001 District Olympiad, 2
Tags: superior algebra , algebra , polynomial
Let $K$ commutative field with $8$ elements. Prove that $(\exists)a\in K$ such that $a^3=a+1$. [i]Mircea Becheanu[/i]

1968 Putnam, A1
Tags: integration , algebra , polynomial , calculus
Prove $ \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx$.

2022 Korea Winter Program Practice Test, 2
Tags: polynomial , inequality , inequalities , algebra
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

1992 IMO Shortlist, 16
Tags: algebra , polynomial
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

1986 India National Olympiad, 5
Tags: polynomial , inequalities , algebra
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

2010 ISI B.Math Entrance Exam, 9
Tags: algebra , polynomial
Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.

2001 Stanford Mathematics Tournament, 14
Tags: college , algebra , polynomial
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.

2013 Saudi Arabia IMO TST, 4
Tags: algebra , polynomial , integer , divisible
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

2008 Bulgaria Team Selection Test, 3
Tags: floor function , ceiling function , induction , algebra , polynomial , combinatorics
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

2022 Costa Rica - Final Round, 2
Tags: algebra , polynomial
Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.

2016 Indonesia TST, 2
Tags: invariant , symmetry , quadratic , algebra , polynomial , vieta , number theory
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2006 Petru Moroșan-Trident, 1
Tags: polynomial , algebra
Prove that the polynom $ X^3-aX-a+1 $ has three integer roots, for an infinite number of integers $ a. $ [i]Liviu Parsan[/i]