Olympiad problems

2003 All-Russian Olympiad, 1

Tags: algebra , polynomial , vieta , area of a triangle , heron's formula , geometry

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

1997 Brazil National Olympiad, 4

Tags: parallelogram , vieta , area of a triangle , heron's formula , geometry , induction

Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence ($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$) Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$

2010 AMC 12/AHSME, 24

Tags: algebra , vieta , inequalities , polynomial

The set of real numbers $ x$ for which \[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\] is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals? $ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$

2021 India National Olympiad, 2

Tags: algebra , cubic , integer root , vieta

Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

2014 AIME Problems, 9

Tags: algebra , vieta , quadratic , polynomial , trigonometry

Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2014 Singapore Senior Math Olympiad, 21

Tags: trigonometry , algebra , quadratic , vieta

Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$

1960 AMC 12/AHSME, 1

Tags: polynomial , vieta , algebra

If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: $ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $

2009 CHKMO, 2

Tags: algebra , polynomial , number theory , euler , vieta , quadratic , function

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

2016 Indonesia TST, 2

Tags: polynomial , invariant , symmetry , quadratic , vieta , number theory , algebra

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2014 Paenza, 3

Tags: polynomial , vieta , algebra

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

2010 Brazil National Olympiad, 1

Tags: algebra , function , vieta , functional equation

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

PEN A Problems, 3

Tags: function , polynomial , vieta , divisibility theory , algebra , induction

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

1977 AMC 12/AHSME, 27

Tags: 3d geometry , quadratic , vieta , geometry , sphere

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is $\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$ $\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$

2012 Online Math Open Problems, 25

Tags: algebra , function , vieta , polynomial

Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]

2019 BMT Spring, 5

Tags: vieta

Find the sum of all real solutions to $ (x^2 - 10x - 12)^{x^2+5x+2} = 1 $

1973 USAMO, 4

Tags: vieta , complex number , polynomial , inequalities , algebra

Determine all roots, real or complex, of the system of simultaneous equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}

1985 IberoAmerican, 1

Tags: symmetry , polynomial , algebra , vieta , modular arithmetic

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

2008 Federal Competition For Advanced Students, Part 2, 2

Tags: algebra , number theory , vieta , polynomial

(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

2017 CMIMC Team, 10

Tags: algebra , vieta , polynomial

The polynomial $P(x) = x^3 - 6x - 2$ has three real roots, $\alpha$, $\beta$, and $\gamma$. Depending on the assignment of the roots, there exist two different quadratics $Q$ such that the graph of $y=Q(x)$ pass through the points $(\alpha,\beta)$, $(\beta,\gamma)$, and $(\gamma,\alpha)$. What is the larger of the two values of $Q(1)$?

1985 IberoAmerican, 3

Tags: algebra , vieta

Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that: \[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]

1961 AMC 12/AHSME, 29

Tags: polynomial , algebra , quadratic , vieta

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is: $ \textbf{(A)}\ x^2-bx-ac=0$ $\qquad\textbf{(B)}\ x^2-bx+ac=0$ $\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$ ${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$ ${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $