Olympiad problems

VMEO III 2006 Shortlist, A9

Tags: polynomial , algebra

Is there any polynomial $P(x)$ with degree $n$ such that $ \underbrace{P(...(P(x))...)}_{m\,\, times \,\, P}$ has all roots from $1,2,..., mn$ ?

2021-IMOC qualification, A2

Tags: algebra , polynomial , functional equation , functional

Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

2023 India Regional Mathematical Olympiad, 3

Tags: algebra , polynomial

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:\\ \[f(a)=bc , f(b)=ac, f(c)=ab\] Dertermine $f(a+b+c)$ in terms of $a,b,c$.

2024 Taiwan TST Round 3, 6

Tags: polynomial , sequence , algebra

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]

1983 IMO, 3

Tags: algebra , polynomial , rearrangement inequality , triangle inequality , inequality

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

1990 Putnam, B5

Tags: algebra , polynomial

Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?

1986 Polish MO Finals, 4

Tags: algebra , polynomial , derivative , functional equation

Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.

1989 IMO Longlists, 51

Tags: polynomial , algebra

Let $ f(x) \equal{} \prod^n_{k\equal{}1} (x \minus{} a_k) \minus{} 2,$ where $ n \geq 3$ and $ a_1, a_2, \ldots,$ an are distinct integers. Suppose that $ f(x) \equal{} g(x)h(x),$ where $ g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $ n \equal{} 3.$

2014 Online Math Open Problems, 13

Tags: algebra , polynomial

Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and \[ g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \] holds for all nonzero real numbers $x$. Find $g(2)$. [i]Proposed by Yang Liu[/i]

2014 Paenza, 3

Tags: algebra , polynomial , vieta

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: algebra , polynomial

Can the equation $x^3 + ax^2 + bx + c = 0$ have only negative roots , if we know that $a+2b+4c=- \frac12 $?

2008 Purple Comet Problems, 7

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

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

PEN Q Problems, 8

Tags: algebra , analytic geometry , graphing lines , slope , combinatorial geometry , polynomial

Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]

2010 Mathcenter Contest, 1

Tags: polynomial , algebra

Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$ and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.$$ [i](tatari/nightmare)[/i]

1985 AMC 12/AHSME, 30

Tags: floor function , algebra , polynomial , product of roots , sum of roots

Let $ \lfloor x \rfloor$ be the greatest integer less than or equal to $ x$. Then the number of real solutions to $ 4x^2 \minus{} 40 \lfloor x \rfloor \plus{} 51 \equal{} 0$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

1998 Harvard-MIT Mathematics Tournament, 4

Tags: function , trigonometry , algebra , polynomial , calculus , derivative

Find the range of $ f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)} $ if $A\neq \dfrac{n\pi}{2}$.

2024 Tuymaada Olympiad, 7

Tags: polynomial , algebra

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n + 1}{n}. \] Prove that $f$ and $g$ have a common root. [i] Proposed by A. Golovanov [/i]

2023 Kurschak Competition, 1

Tags: number theory , polynomial , algebra

Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.

2006 Iran MO (3rd Round), 4

Tags: polynomial , algebra

$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$

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)

1999 China National Olympiad, 2

Tags: polynomial , function , induction , algebra

Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$. a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$. b) Find an explicit expression for $f_n(x)$.

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , polynomial , algebra

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

2021 Purple Comet Problems, 28

Tags: algebra , polynomial

Let $z_1$, $z_2$, $z_3$, $\cdots$, $z_{2021}$ be the roots of the polynomial $z^{2021}+z-1$. Evaluate $$\frac{z_1^3}{z_{1}+1}+\frac{z_2^3}{z_{2}+1}+\frac{z_3^3}{z_{3}+1}+\cdots+\frac{z_{2021}^3}{z_{2021}+1}.$$

2019 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial

The real numbers $a$, $b$ and $c$ verify that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct real roots; these roots are equal to $\tan y$, $\tan 2y$ and $\tan 3y$, for some real number $y$. Find all possible values of $y$, $0\leq y < \pi$.

Russian TST 2018, P3

Tags: algebra , polynomial

Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.