Olympiad problems

III Soros Olympiad 1996 - 97 (Russia), 11.7

Tags: polynomial , algebra

Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

1980 IMO, 1

Tags: number theory , greatest common divisor , algebra , polynomial

Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.

2013 Hanoi Open Mathematics Competitions, 11

Tags: minimum , maximum , polynomial

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

2003 Singapore Senior Math Olympiad, 2

Tags: algebra , binomial , combination , polynomial

For each positive integer $k$, we define the polynomial $S_k(x)=1+x+x^2+x^3+...+x^{k-1}$ Show that $n \choose 1$ $S_1(x) +$ $n \choose 2$ $S_2(x) +$ $n \choose 3$ $S_3(x)+...+$ $n \choose n$ $S_n(x) = 2^{n-1}S_n\left(\frac{1+x}{2}\right)$ for every positive integer $n$ and every real number $x$.

1998 Poland - First Round, 4

Tags: strong induction , algebra , polynomial , induction

Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.

2015 Taiwan TST Round 3, 2

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]

2015 Baltic Way, 16

Tags: polynomial , number theory , algebra

Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]

2015 AIME Problems, 14

Tags: polynomial , algebra

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

2016 Iran MO (3rd Round), 3

Tags: polynomial , algebra

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

2009 Romania Team Selection Test, 2

Tags: polynomial , algebra

Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$.

2018 Iran Team Selection Test, 3

Tags: integer polynomial , number theory , polynomial

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]

2019 Vietnam National Olympiad, Day 2

Tags: factoring polynomials , polynomial , algebra

Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

PEN Q Problems, 11

Tags: polynomial , algebra

Show that the polynomial $x^{8} +98 x^{4}+1$ can be expressed as the product of two nonconstant polynomials with integer coefficients.

1980 IMO, 19

Tags: diophantine equation , polynomial , algebra , equation , number theory

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2015 Iran Team Selection Test, 5

Tags: algebra , polynomial

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

1962 AMC 12/AHSME, 9

Tags: polynomial , calculus , algebra , integration

When $ x^9\minus{}x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: $ \textbf{(A)}\ \text{more than 5} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

2011 AIME Problems, 9

Tags: logarithm , rational root theorem , algebra , trigonometry , polynomial

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

1955 AMC 12/AHSME, 25

Tags: algebra , polynomial

One of the factors of $ x^4\plus{}2x^2\plus{}9$ is: $ \textbf{(A)}\ x^2\plus{}3 \qquad \textbf{(B)}\ x\plus{}1 \qquad \textbf{(C)}\ x^2\minus{}3 \qquad \textbf{(D)}\ x^2\minus{}2x\minus{}3 \qquad \textbf{(E)}\ \text{none of these}$

1968 Putnam, A1

Tags: algebra , calculus , integration , polynomial

Prove $ \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx$.

2014 USA TSTST, 3

Tags: induction , derivative , polynomial , trigonometry , calculus , inequalities , algebra

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

1998 Romania Team Selection Test, 2

Tags: algebra , polynomial , functional equation

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

1980 IMO Longlists, 4

Tags: convex polygon , polynomial , circumcircle , geometry , algebra

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

2012 ELMO Shortlist, 6

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

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2011 NIMO Summer Contest, 7

Tags: greatest common divisor , algebra , polynomial , number theory

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

2020 USA IMO Team Selection Test, 5

Tags: polynomial , integer polynomial , inequalities , number theory

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]