This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 236

2015 AMC 12/AHSME, 12
Tags: algebra , polynomial , Vieta , AMC
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? $ \textbf {(A) } 15 \qquad \textbf {(B) } 15.5 \qquad \textbf {(C) } 16 \qquad \textbf {(D) } 16.5 \qquad \textbf {(E) } 17 $

1997 China Team Selection Test, 1
Tags: algebra , polynomial , inequalities , Vieta , induction , algebra unsolved , Polynomials
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

2012 Math Prize For Girls Problems, 20
Tags: trigonometry , algebra , polynomial , Vieta
There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation \[ \tan(15 x) = 15 \tan(x) . \] Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum \[ \frac{1}{\tan^2 r_1} + \frac{1}{\tan^2 r_2} + \frac{1}{\tan^2 r_3} + \frac{1}{\tan^2 r_4} + \frac{1}{\tan^2 r_5} + \frac{1}{\tan^2 r_6} \, ? \]

2013 NIMO Summer Contest, 4
Tags: algebra , polynomial , Vieta
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

2003 AMC 10, 5
Tags: Vieta , quadratics , algebra , polynomial
Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$? $ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

MathLinks Contest 7th, 5.1
Tags: algebra , polynomial , inequalities , Vieta , product of roots , absolute value
Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

2016 Indonesia TST, 2
Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

1998 All-Russian Olympiad, 1
Tags: conics , parabola , Support , algebra , polynomial , Vieta , geometry
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.

1991 AMC 12/AHSME, 20
Tags: geometry , algebra , polynomial , Vieta , logarithms , AMC
The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

2012 Stanford Mathematics Tournament, 6
Tags: algebra , polynomial , Vieta , sum of roots
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 $f(0)^3+g(0)^3$

PEN H Problems, 9
Tags: algebra , polynomial , Vieta , LaTeX , absolute value , Diophantine Equations
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.

2008 AMC 10, 9
Tags: quadratics , Vieta , algebra , AMC
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac{b}{a} \qquad \textbf{(D)}\ \frac{2b}{a} \qquad \textbf{(E)}\ \sqrt{2b\minus{}a}$

2018-2019 Winter SDPC, 1
Tags: algebra , polynomial , Vieta
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

2011 NIMO Problems, 9
Tags: algebra , polynomial , Vieta
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

2008 India Regional Mathematical Olympiad, 3
Tags: inequalities , Vieta , calculus , derivative , trigonometry , LaTeX , algebra
Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that: \[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3} \] [19 points out of 100 for the 6 problems]

2009 USA Team Selection Test, 5
Tags: quadratics , inequalities , algebra , polynomial , Vieta , number theory , USA TST
Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

2014 Contests, 1
Tags: Vieta
If $\alpha$ and $\beta$ are the roots of the equation $3x^2+x-1=0$, where $\alpha>\beta$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$. $ \textbf{(A) }\frac{7}{9}\qquad\textbf{(B) }-\frac{7}{9}\qquad\textbf{(C) }\frac{7}{3}\qquad\textbf{(D) }-\frac{7}{3}\qquad\textbf{(E) }-\frac{1}{9} $

1996 Canada National Olympiad, 1
Tags: algebra , polynomial , Vieta , algebra unsolved
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

2006 Putnam, A5
Tags: Putnam , trigonometry , algebra , polynomial , ratio , induction , Vieta
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.

2007 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , algebra , polynomial , Vieta , binomial theorem
Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

2008 AIME Problems, 7
Tags: Vieta , AMC , AIME , algebra , polynomial , AIME I
Let $ r$, $ s$, and $ t$ be the three roots of the equation \[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.

2020 HK IMO Preliminary Selection Contest, 11
Tags: algebra , polynomial , Vieta
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

1991 AIME Problems, 1
Tags: AMC , AIME , algebra , polynomial , Vieta
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]

2007 Irish Math Olympiad, 1
Tags: algebra , polynomial , Vieta , algebra unsolved
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$ Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.

2015 CCA Math Bonanza, TB2
Tags: algebra , polynomial , Vieta
If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$. [i]2015 CCA Math Bonanza Tiebreaker Round #2[/i]