Olympiad problems

2003 Federal Competition For Advanced Students, Part 2, 1

Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

1955 AMC 12/AHSME, 25

Tags: polynomial , algebra

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}$

1988 Czech And Slovak Olympiad IIIA, 2

Tags: algebra , polynomial

If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.

2015 Postal Coaching, 3

Tags: algebra , polynomial

Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$

2008 Korea - Final Round, 2

Tags: polynomial , number theory , relatively prime , algebra

Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b \mid f(a)+f(b)$.

2012 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt , algebra , polynomial , function , trigonometry

How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?

1978 Romania Team Selection Test, 3

Tags: algebra , polynomial , analytic geometry , geometry

Let $ P[X,Y] $ be a polynomial of degree at most $ 2 .$ If $ A,B,C,A',B',C' $ are distinct roots of $ P $ such that $ A,B,C $ are not collinear and $ A',B',C' $ lie on the lines $ BC,CA, $ respectively, $ AB, $ in the planar representation of these points, show that $ P=0. $

2021 Ecuador NMO (OMEC), 2

Tags: algebra , polynomial

Let $P(x)$ a grade 3 polynomial such that: $$P(1)=1, P(2)=4, P(3)=9$$ Find the value of $P(10)+P(-6)$

1986 Traian Lălescu, 2.1

Tags: binom , determinant , matrices , polynomial , linear algebra

Show that for any natural numbers $ m,n\ge 3, $ the equation $ \Delta_n (x)=0 $ has exactly two distinct solutions, where $$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$

2023 Indonesia TST, N

Tags: number theory , polynomial

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

2013 Miklós Schweitzer, 2

Tags: algebra , polynomial , vieta , number theory

Prove there exists a constant $k_0$ such that for any $k\ge k_0$, the equation \[a^{2n}+b^{4n}+2013=ka^nb^{2n}\] has no positive integer solutions $a,b,n$. [i]Proposed by István Pink.[/i]

2015 Saudi Arabia Pre-TST, 1.2

Tags: polynomial , algebra

How many polynomials $P$ of integer coefficients and degree at most $4$ satisfy $0 \le P(x) < 72$ for all $x\in \{0, 1, 2, 3, 4\}$? Harvard-MIT Mathematics Tournament 2011

1996 All-Russian Olympiad, 4

Tags: polynomial , algebra

Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

2015 Indonesia MO Shortlist, A2

Tags: natural number , polynomial , functional equation , algebra

Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.

1980 Dutch Mathematical Olympiad, 1

Tags: algebra , polynomial , inequalities

$f(x) = x^3-ax+1$ , $a \in R$ has three different zeros in $R$. Prove that for the zero $x_o$ with the smallest absolute value holds: $\frac{1}{a}< x_0 < \frac{2}{a}$

2013 NIMO Problems, 10

Tags: algebra , polynomial

Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$. [i]Proposed by Evan Chen[/i]

2023 SAFEST Olympiad, 6

Tags: algebra , polynomial

Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$ [i]Proposed by Navid Safaei, Iran[/i]

2018 Purple Comet Problems, 18

Tags: complex number , polynomial

Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.

1976 Miklós Schweitzer, 4

Tags: calculus , integration , algebra , function , domain , polynomial , ring theory

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

2018 Mathematical Talent Reward Programme, SAQ: P 2

Tags: polynomial , algebra

$P(x)$ is polynomial with real coefficients such that $\forall n \in \mathbb{Z}, P(n) \in \mathbb{Z}$. Prove that every coefficients of $P(x)$ is rational numbers.

2016 Tournament Of Towns, 6

Tags: combinatorial game , algebra , polynomial , number theory , game theory , combinatorics

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

2007 Romania Team Selection Test, 4

Tags: algebra , polynomial , modular arithmetic , induction , number theory

i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]

2015 Romania National Olympiad, 2

Tags: function , algebra , polynomial , quadratic

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

2023 Bangladesh Mathematical Olympiad, P10

Tags: algebra , polynomial , inequalities

Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$, where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$

2018 China Team Selection Test, 5

Tags: algebra , polynomial

Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$