Olympiad problems

2014 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , polynomial , hmmt , vieta , sum of roots

Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.

1992 India National Olympiad, 6

Tags: polynomial , integration , calculus , algebra

Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.

2001 China Team Selection Test, 3

Tags: polynomial , algebra , function , functional equation

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

2017 Vietnam National Olympiad, 2

Tags: algebra , polynomial

Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} \]

2023 Belarusian National Olympiad, 9.5

Tags: algebra , polynomial

The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

2018 IMAR Test, 2

Tags: polynomial , minimum value , permutation , algebra

Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum \[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers. [i]Fedor Petrov[/i]

2019 Taiwan TST Round 3, 2

Tags: algebra , polynomial

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

2006 District Olympiad, 3

Tags: algebra , polynomial , superior algebra

Prove that if $A$ is a commutative finite ring with at least two elements and $n$ is a positive integer, then there exists a polynomial of degree $n$ with coefficients in $A$ which does not have any roots in $A$.

PEN G Problems, 16

Tags: inequalities , algebra , integration , irrational number , polynomial

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

1992 IMO Shortlist, 16

Tags: algebra , polynomial

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

2023 AIME, 9

Tags: algebra , polynomial

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.

2009 AMC 12/AHSME, 19

Tags: polynomial , algebra , search , vieta , prime number , number theory

For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

2003 India IMO Training Camp, 7

Tags: number theory , limit , polynomial , algebra

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2020 JBMO Shortlist, 1

Tags: polynomial , system of equations , algebra

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

CVM 2020, Problem 6

Tags: polynomial , algebra

Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$? $\textbf{6.1.}$ A monic polynomial is one that has a main coefficient equal to $1$. For example, the polynomial $P(x) = x^3 + 5x^2 - 3x + 7$ is a monic polynomial [i]Proposed by Lenin Vasquez, Copan[/i]

1982 IMO Longlists, 56

Tags: graphing lines , analytic geometry , inequalities , algebra , conic , parabola , polynomial

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$, [b](a)[/b] $|f(x)| \leq 5/4$, [b](b)[/b] $|g(x)| \leq 2$.

2025 Alborz Mathematical Olympiad, P2

Tags: polynomial , positive integer , positive coefficients , algebra

Suppose that for polynomials $ P, Q, R $ with positive integer coefficients, the following two conditions hold: $\bullet$ The constant terms of $ P, Q, R $ are equal. $\bullet$ For all real numbers $ x $, the following relations hold: \[ P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))). \] Prove that for every real number $ x $, $ P(x) = Q(x) = R(x) $. Proposed by Soroush Behroozifar & Ali Nazarboland

2019 AMC 12/AHSME, 14

Tags: complex number , algebra , polynomial

For a certain complex number $c$, the polynomial \[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\] has exactly 4 distinct roots. What is $|c|$? $\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$

2010 ELMO Problems, 2

Tags: polynomial , algebra , floor function , number theory

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

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? $\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

1997 German National Olympiad, 6a

Tags: prime , algebra , divide , polynomial

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

2012 Czech-Polish-Slovak Match, 2

Tags: algebra , function , polynomial

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x+f(y))-f(x)=(x+f(y))^4-x^4\] for all $x,y \in \mathbb{R}$.

2019-2020 Fall SDPC, 3

Tags: polynomial , algebra

Find all polynomials $P$ with integer coefficients such that for all positive integers $x,y$, $$\frac{P(x)-P(y)}{x^2+y^2}$$ evaluates to an integer (in particular, it can be zero).

1992 IMO Shortlist, 1

Tags: algebra , polynomial , vieta , quadratic , number theory

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that [i](i)[/i] $ x$ and $ y$ are relatively prime; [i](ii)[/i] $ y$ divides $ x^2 \plus{} m$; [i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$ [i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)

2005 Morocco TST, 3

Tags: algebra , polynomial , inequalities

The real numbers $a_1,a_2,...,a_{100}$ satisfy the relationship : $a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101$ Prove that $|a_k| \leq 10$ for all $k \in \{1,2,...,100\}$