Olympiad problems

2001 Romania National Olympiad, 4

Tags: polynomial , algebra

Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.

PEN C Problems, 6

Tags: quadratic residue , algebra , polynomial , absolute value

Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

1983 AIME Problems, 3

Tags: polynomial , vieta , function , domain , quadratic , algebra

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

2004 All-Russian Olympiad, 3

Tags: algebra , polynomial , calculus

The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) \equal{} S(Q(x)) \quad \forall x.$

2011 All-Russian Olympiad, 2

Tags: quadratic , invariant , vieta , induction , algebra , polynomial , arithmetic sequence

In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

Kvant 2025, M2828

Tags: algebra , polynomial

Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps? [i]M. Didin[/i]

1987 Austrian-Polish Competition, 2

Tags: prime divisor , polynomial , algebra

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.

1999 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

2009 USA Team Selection Test, 5

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

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

2006 Junior Balkan Team Selection Tests - Moldova, 3

Tags: polynomial , algebra

Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.8

Tags: polynomial , algebra , periodic

A polynomial with rational coefficients is called [i]integer[/i], if it takes integer values for all integer values of the variable. For an integer polynomial $P$, consider the sequence $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$ a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer $k$ and for all natural $i$, the $i$-th and ($i+2^k$)th members of the sequence are equal). b) Prove that for any periodic sequence consisting of $(- 1)$ and $ 1$ and with a period of some power of two, there exists a integer, polynomial P for which this sequence is $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$

2016 Hanoi Open Mathematics Competitions, 15

Tags: polynomial , integer , algebra , prime

Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

2006 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , algebra , polynomial

Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]

1982 AMC 12/AHSME, 1

Tags: algebra , polynomial

When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is $\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$

2024 USA TSTST, 2

Tags: algebra , polynomial , continued fraction

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and \[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \] Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$. [i]Andrew Gu[/i]

1974 Czech and Slovak Olympiad III A, 4

Tags: algebra , polynomial , function , bounded , set

Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.

2015 Belarus Team Selection Test, 3

Tags: polynomial , functional inequality , algebra

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

2023 Tuymaada Olympiad, 5

Tags: analytic geometry , polynomial , algebra

A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.

1989 USAMO, 3

Tags: polynomial , inequalities , algebra

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

2009 IMS, 4

Tags: ratio , vector , algebra , polynomial , limit , induction , absolute value

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: abstract algebra , ring theory , polynomial , polynomial ring

Let be a ring $ R $ which has the property that there exist two distinct natural numbers $ s,t $ such that for any element $ x $ of $ R, $ the equation $ x^s=x^t $ is true. Show that there exists a polynom in $ R[X] $ of degree $$ |s-t|\left( 1+|s-t| \right) $$ such that all the elements of $ R $ are roots of it.

KoMaL A Problems 2020/2021, A. 801

Tags: algebra , polynomial

For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]

1993 Irish Math Olympiad, 1

Tags: polynomial , algebra

The following is known about the reals $ \alpha$ and $ \beta$ $ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$ Determine $ \alpha+\beta$

2004 China Team Selection Test, 3

Tags: algebra , polynomial , calculus , integration , number theory

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2023 Miklós Schweitzer, 8

Tags: complex analysis , polynomial

Let $q{}$ be an arbitrary polynomial with complex coefficients which is not identically $0$ and $\Gamma_q =\{z : |q(z)| = 1\}$ be its contour line. Prove that for every point $z_0\in\Gamma_q$ there is a polynomial $p{}$ for which $|p(z_0)| = 1$ and $|p(z)|<1$ for any $z\in\Gamma_q\setminus\{z_0\}.$