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

1967 IMO Shortlist, 5
Tags: linear algebra , matrix , algebra , system of equations , polynomial
Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

2015 India Regional MathematicaI Olympiad, 2
Tags: integer polynomial , even , polynomial , algebra
Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.

2019 PUMaC Algebra A, 3
Tags: polynomial , algebra , quadratic
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

1990 Romania Team Selection Test, 3
Tags: functional equation , functional , algebra , polynomial
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2007 Today's Calculation Of Integral, 200
Tags: abstract algebra , polynomial , integration , algebra , trigonometry , calculus , function
Evaluate the following definite integral. \[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]

2000 Vietnam National Olympiad, 3
Tags: polynomial , complex number , algebra
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.

2009 Postal Coaching, 3
Tags: algebra , polynomial
Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$, $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$.]

2020 IMC, 4
Tags: algebra , polynomial
A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$

2017 India IMO Training Camp, 1
Tags: algebra , polynomial
Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$. (a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots. (b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.

2007 Princeton University Math Competition, 7
Tags: polynomial , limit , algebra , linear algebra , matrix , vector
Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$, \[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\] find $\lim_{n \to \infty} \frac{x_n}{y_n}$.

1985 Traian Lălescu, 1.2
Tags: algebra , polynomial
Prove that all real roots of the polynomial $$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$ are positive.

2012 Korea - Final Round, 3
Tags: algebra , combinatorics , ring theory , analytic geometry , polynomial
Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

2005 MOP Homework, 4
Tags: algebra , derivative , calculus , combinatorics , polynomial
Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.

1988 IMO Shortlist, 2
Tags: polynomial , algebra , generating functions , binomial coefficient , function
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

PEN N Problems, 14
Tags: algebra , arithmetic sequence , more sequences , polynomial
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

2007 IMS, 6
Tags: algebra , superior algebra , ring theory , polynomial
Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.

1993 IMO Shortlist, 7
Tags: algebra , irreducible , factoring polynomials , irreducible polynomial , polynomial
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

1986 Austrian-Polish Competition, 2
Tags: algebra , conic , polynomial
The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$

1978 IMO Longlists, 6
Tags: geometry , polynomial , function , trigonometry , algebra
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

2016 Latvia Baltic Way TST, 3
Tags: algebra , polynomial
Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers: $$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$

2012 Thailand Mathematical Olympiad, 9
Tags: polynomial , real root , algebra , inequalities
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.

2022 Bulgarian Spring Math Competition, Problem 12.3
Tags: algebra , polynomial
Let $P,Q\in\mathbb{R}[x]$, such that $Q$ is a $2021$-degree polynomial and let $a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022}$ be real numbers such that $a_{1}a_{2}\ldots a_{2022}\neq 0$. If for all real $x$ \[P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022})\] prove that $P(x)$ has a real root.

2018 Canada National Olympiad, 4
Tags: number theory , polynomial , algebra
Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

2018-2019 Winter SDPC, 1
Tags: vieta , algebra , polynomial
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

1973 IMO Shortlist, 11
Tags: algebra , root , minimum value , coefficient , polynomial
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.