Olympiad problems

1995 IMC, 9

Let all roots of an $n$-th degree polynomial $P(z)$ with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial $$2zP'(z)-nP(z)$$ lie on the same circle.

2018 Azerbaijan Senior NMO, 2

Tags: algebra , polynomial

$P(x)$ is a fifth degree polynomial. $P(2018)=1$, $P(2019)=2$ $P(2020)=3$, $P(2021)=4$, $P(2022)=5$. $P(2017)=?$

2023 Stars of Mathematics, 4

Tags: algebra , polynomial

Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]

2017 CHMMC (Fall), 10

Tags: algebra , polynomial

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?

2008 Alexandru Myller, 2

Tags: algebra , polynomial

Find all natural numbers $ n\ge 3 $ and real numbers $ a $ which have the property that the polynomial $ X^n-aX-1 $ admits a monic quadratic integer polynomial. [i]Mihai Bălună[/i]

1998 Brazil Team Selection Test, Problem 4

Tags: algebra , polynomial

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

2010 Moldova Team Selection Test, 1

Tags: polynomial , algebra

Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$

1976 AMC 12/AHSME, 7

Tags: function , algebra , polynomial

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if $\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$ $\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

2020 USA TSTST, 7

Tags: algebra , polynomial , complex number

Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1. [i]Ankan Bhattacharya[/i]

1951 Miklós Schweitzer, 10

Tags: algebra , polynomial , modular arithmetic , ring theory

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

2006 Princeton University Math Competition, 4

Tags: algebra , polynomial

Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.

2013 Korea National Olympiad, 3

Tags: algebra , polynomial , quadratic , induction , calculus , integration , modular arithmetic

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

1966 IMO Longlists, 35

Tags: algebra , polynomial , irrational number , root

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

1974 Miklós Schweitzer, 7

Tags: algebra , polynomial , trigonometry , integration , real analysis

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

2005 AIME Problems, 6

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

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

2020 Azerbaijan IMO TST, 2

Tags: number theory , polynomial

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

2010 AMC 12/AHSME, 6

Tags: algebra , polynomial , vieta

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

2019 Moldova Team Selection Test, 7

Tags: algebra , polynomial

Let $P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0$ be a polynomial with all positive coefficients. Prove that there exists a permutation $(b_{2n+1},b_{2n},...,b_1,b_0)$ of numbers $(a_{2n+1},a_{2n},...,a_1,a_0)$ such that the polynomial $Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0$ has exactly one real root.

2019 Korea Junior Math Olympiad., 5

Tags: number theory , prime number , algebra , polynomial

For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.

2023 Indonesia TST, A

Tags: polynomial , algebra , integer polynomial

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

2019-IMOC, A2

Tags: algebra , polynomial

Given a real number $t\ge3$, suppose a polynomial $f\in\mathbb R[x]$ satisfies $$\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.$$Prove that $\deg f\ge n$.

2005 Brazil Undergrad MO, 1

Tags: linear algebra , matrix , algebra , polynomial , function

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

2010 Math Prize For Girls Problems, 20

Tags: trigonometry , algebra , polynomial , vieta , function , ratio , complex number

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

2012 AMC 12/AHSME, 20

Tags: algebra , polynomial

Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $

2010 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra , polynomial

$f(x)$ is square trinomial. Is it always possible to find polynomial $g(x)$ with fourth degree, such that $f(g(x))=0$ has not roots?