Olympiad problems

1998 Italy TST, 4

Tags: algebra , polynomial

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

2022 Singapore MO Open, Q5

Tags: number theory , polynomial , modulo , prime number , algebra

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

2009 India National Olympiad, 6

Tags: quadratic , algebra , polynomial , inequality

Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that: $ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.

2018 CHMMC (Fall), 5

Tags: algebra , polynomial

Let $a,b, c, d,e$ be the roots of $p(x) = 2x^5 - 3x^3 + 2x -7$. Find the value of $$(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).$$

2005 AMC 12/AHSME, 24

Tags: algebra , polynomial , quadratic , function

Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$? $ \textbf{(A)}\ 19\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 32$

2011 Putnam, A4

Tags: linear algebra , matrix , modular arithmetic , vector , polynomial , putnam matrices

For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

1999 AIME Problems, 3

Tags: algebra , polynomial , vieta , quadratic , special factorizations

Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.

2023 Miklós Schweitzer, 9

Tags: polynomial , analysis , functional analysis , real analysis

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

2003 China Team Selection Test, 3

Tags: polynomial , inequalities , algebra

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

2006 District Olympiad, 3

Tags: superior algebra , polynomial , algebra

Prove that if $A$ is a commutative finite ring with at least two elements and $n$ is a positive integer, then there exists a polynomial of degree $n$ with coefficients in $A$ which does not have any roots in $A$.

1978 Chisinau City MO, 160

Tags: factor , polynomial , algebra

Factor the polynomial $P (x) = 1 + x +x^2+...+x^{2^k-1}$

2004 Romania Team Selection Test, 4

Tags: geometry , circumcircle , limit , gauss , algebra , polynomial , logarithm

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

2003 Swedish Mathematical Competition, 4

Tags: algebra , polynomial

Determine all polynomials $P$ with real coeffients such that $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$ for all real $x$.

1987 AMC 12/AHSME, 28

Tags: algebra , polynomial , complex number

Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ -a \qquad\textbf{(E)}\ -b $

1993 Greece National Olympiad, 4

Tags: symmetry , algebra , polynomial , linear algebra , matrix , inequalities

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

2013 Kosovo National Mathematical Olympiad, 2

Tags: algebra , polynomial

Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of math teacher?

2014 USA Team Selection Test, 1

Tags: algebra , polynomial

Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying \[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \] Prove that \[ 2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2 \] has no real roots.

2019 IFYM, Sozopol, 8

Tags: algebra , polynomial , divisibility , number theory

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

2005 China Western Mathematical Olympiad, 1

Tags: polynomial , algebra

It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial.

2023 Germany Team Selection Test, 3

Tags: polynomial , number theory , prime number , algebra

Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]

2010 Junior Balkan MO, 1

Tags: polynomial , function , algebra

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

2007 Iran Team Selection Test, 2

Tags: algebra , polynomial , search , number theory

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

1995 Vietnam National Olympiad, 2

Tags: polynomial , algebra

Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.

2012 Romania National Olympiad, 4

Tags: polynomial , superior algebra , algebra

[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients. [/color]

PEN F Problems, 8

Tags: rational number , algebra , polynomial , analytic geometry , function , induction

Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.