Olympiad problems

2012 India Regional Mathematical Olympiad, 2

Tags: sum of roots , vieta , binomial theorem , inequalities , algebra , polynomial

Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

2009 Harvard-MIT Mathematics Tournament, 8

Tags: vieta , sum of roots , algebra , polynomial

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

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 $f(0)^3+g(0)^3$

2021 Alibaba Global Math Competition, 19

Tags: nested radical , algebra , roots of unity , sum of roots , polynomial

Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.

1985 AMC 12/AHSME, 30

Tags: algebra , polynomial , product of roots , sum of roots , floor function

Let $ \lfloor x \rfloor$ be the greatest integer less than or equal to $ x$. Then the number of real solutions to $ 4x^2 \minus{} 40 \lfloor x \rfloor \plus{} 51 \equal{} 0$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2022 MMATHS, 9

Tags: polynomial , sum of roots , algebra

Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.

2011 Purple Comet Problems, 9

Tags: polynomial , quadratic , quadratic formula , sum of roots , product of roots , algebra

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

2012 Tuymaada Olympiad, 2

Tags: polynomial , quadratic , sum of roots , inequalities , algebra

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$. [i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]

1978 IMO Longlists, 3

Tags: sum of roots , polynomial , algebra , floor function

Find all numbers $\alpha$ for which the equation \[x^2 - 2x[x] + x -\alpha = 0\] has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)

2016 Mathematical Talent Reward Programme, MCQ: P 1

Tags: trigonometric equation , sum of roots , polynomial , algebra

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is [list=1] [*] $\frac{\pi}{2}$ [*] 0 [*] 1 [*] None of these [/list]

IV Soros Olympiad 1997 - 98 (Russia), 9.9

Tags: sum of roots , number theory

Find an odd natural number not exceeding $1000$ if you know that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.

2010 Contests, 2

Tags: number theory , real analysis , quadratic , complex number , sum of roots , algebra , polynomial

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2011 Israel National Olympiad, 2

Tags: algebra , root , sum of roots , polynomial

Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.

2006 Romania Team Selection Test, 2

Tags: polynomial , real analysis , number theory , quadratic , complex number , sum of roots , algebra

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2004 Austria Beginners' Competition, 3

Tags: quadratic , sum of roots , polynomial , algebra

Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.

2014 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , polynomial , hmmt , vieta , sum of roots

Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.

1987 Federal Competition For Advanced Students, P2, 6

Tags: algebra , sum of roots , polynomial

Determine all polynomials $ P_n(x)\equal{}x^n\plus{}a_1 x^{n\minus{}1}\plus{}...\plus{}a_{n\minus{}1} x\plus{}a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).

1996 Canadian Open Math Challenge, 1

Tags: polynomial , algebra , quadratic , quadratic formula , sum of roots , product of roots , rational root theorem

The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?

1998 India National Olympiad, 5

Tags: inequalities , algebra , polynomial , quadratic , sum of roots , area of a triangle

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

2020 BMT Fall, 2

Tags: polynomial , algebra , sum of roots

Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$.

2019 Ecuador NMO (OMEC), 4

Tags: combinatorics , game theory , sum of roots , strategy

Let $n> 1$ be a positive integer. Danielle chooses a number $N$ of $n$ digits but does not tell her students and they must find the sum of the digits of $N$. To achieve this, each student chooses and says once a number of $n$ digits to Danielle and she tells how many digits are in the correct location compared with $N$. Find the minimum number of students that must be in the class to ensure that students have a strategy to correctly find the sum of the digits of $N$ in any case and show a strategy in that case.