Olympiad problems

2012 ISI Entrance Examination, 4

Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.

2019 IMO Shortlist, A6

Tags: polynomial , algebra

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

2013 ELMO Shortlist, 7

Tags: algebra , number theory , polynomial , modular arithmetic

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2014 Canadian Mathematical Olympiad Qualification, 5

Tags: algebra , irreducibility , polynomial

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

2013 Harvard-MIT Mathematics Tournament, 20

Tags: hmmt , complex number , polynomial , algebra

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

2012 Iran MO (3rd Round), 4

Tags: polynomial , algebra

Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$ \[x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|\] and the following polynomial has only one positive real root like $\beta$ \[x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.\] And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.

2018 Iran MO (3rd Round), 4

Tags: algebra , polynomial

Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.

2006 Vietnam National Olympiad, 5

Tags: polynomial , algebra

Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: \[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \]

2018 Germany Team Selection Test, 1

Tags: ineqstd , polynomial , algebra

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.

2010 N.N. Mihăileanu Individual, 1

Tags: conic , quadratic , algebra , polynomial

Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.

2005 AIME Problems, 6

Tags: quadratic , function , algebra , floor function , polynomial , trigonometry

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.

2018 Estonia Team Selection Test, 12

Tags: divisible , polynomial

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

2012 ELMO Shortlist, 9

Tags: function , polynomial , inequalities , algebra , induction

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

2021 Saudi Arabia Training Tests, 34

Tags: number theory , divide , polynomial

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

2017 China Team Selection Test, 2

Tags: algebra , polynomial , abstract algebra

Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.

2022 Junior Macedonian Mathematical Olympiad, P5

Tags: number theory , polynomial

Let $n$ be a positive integer such that $n^5+n^3+2n^2+2n+2$ is a perfect cube. Prove that $2n^2+n+2$ is not a perfect cube. [i]Proposed by Anastasija Trajanova[/i]

1977 USAMO, 3

Tags: polynomial , algebra

If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.

2005 MOP Homework, 2

Tags: polynomial , algebra

Determine if there exist four polynomials such that the sum of any three of them has a real root while the sum of any two of them does not.

2009 China Team Selection Test, 2

Tags: algebra , polynomial

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

2001 District Olympiad, 3

Tags: algebra , real analysis , function , integration , polynomial , calculus

Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

2024 Bulgarian Autumn Math Competition, 10.3

Tags: prime divisor , polynomial , floor function , number theory , algebra

Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.

1970 IMO Longlists, 54

Tags: polynomial , algebra

Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$

2012 Stanford Mathematics Tournament, 6

Tags: vieta , sum of roots , algebra , polynomial

There exist two triples of real numbers $(a,b,c)$ such that $a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}$ are the roots to the cubic equation $x^3-5x^2-15x+3$ listed in increasing order. Denote those $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$. If $a_1$, $b_1$, and $c_1$ are the roots to monic cubic polynomial $f$ and $a_2, b_2$, and $c_2$ are the roots to monic cubic polynomial $g$, ﬁnd $f(0)^3+g(0)^3$

1973 Polish MO Finals, 1

Tags: polynomial , algebra

Prove that every polynomial is a difference of two increasing polynomials.

1971 IMO Shortlist, 8

Tags: root , polynomial , equation , algebra

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$