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 10, 16
Tags: vieta , algebra , polynomial , symmetry , rational root theorem , system of equations
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $

2012 NIMO Problems, 2
Tags: algebra , polynomial , vieta
If $r_1$, $r_2$, and $r_3$ are the solutions to the equation $x^3 - 5x^2 + 6x - 1 = 0$, then what is the value of $r_1^2 + r_2^2 + r_3^2$? [i]Proposed by Eugene Chen[/i]

2015 CCA Math Bonanza, TB2
Tags: polynomial , vieta , algebra
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]

2009 AMC 12/AHSME, 19
Tags: algebra , polynomial , search , vieta , prime number , number theory
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

1999 National Olympiad First Round, 34
Tags: modular arithmetic , algebra , polynomial , vieta
For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

2003 Romania Team Selection Test, 7
Tags: polynomial , vieta , trigonometry , algebra
Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$. [i]Laurentiu Panaitopol[/i]

2007 Harvard-MIT Mathematics Tournament, 20
Tags: algebra , polynomial , vieta
For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

1997 Brazil National Olympiad, 4
Tags: vieta , parallelogram , induction , area of a triangle , heron's formula , geometry
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 $1/2$

1971 IMO Shortlist, 3
Tags: polynomial , algebra , system of equations , vieta
Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

2013 AMC 12/AHSME, 22
Tags: vieta , logarithm
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation \[8(\log_n x)(\log_m x) - 7 \log_n x - 6 \log_m x - 2013 = 0\] is the smallest possible integer. What is $m+n$? ${ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272 $

1977 AMC 12/AHSME, 27
Tags: geometry , 3d geometry , sphere , quadratic , vieta
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 $\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$ $\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$

PEN A Problems, 1
Tags: quadratic , vieta , algebra , ratio , number theory , relatively prime
Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

2014 Singapore Senior Math Olympiad, 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} $

2006 Putnam, A5
Tags: 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.

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.

2006 Pan African, 2
Tags: vieta , algebra , polynomial , rational root theorem , number theory
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.

2002 AMC 10, 11
Tags: quadratic , vieta , algebra , polynomial
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$. $\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$

1965 AMC 12/AHSME, 22
Tags: algebra , polynomial , vieta
If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds: $ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$ $ \textbf{(B)}\ \text{for all values of }x$ $ \textbf{(C)}\ \text{only when }x \equal{} 0$ $ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$ $ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$

2008 Harvard-MIT Mathematics Tournament, 5
Tags: algebra , polynomial , vieta , function
Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.

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 $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

2015 AMC 10, 23
Tags: quadratic , function , polynomial , vieta , sfft , algebra
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2015 AMC 10, 12
Tags: quadratic , algebra , polynomial , vieta , quadratic formula , absolute value
Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

PEN A Problems, 5
Tags: hyperbola , symmetry , vieta , number theory , conic , algebra
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

2015 AMC 12/AHSME, 18
Tags: quadratic , function , polynomial , vieta , sfft , algebra
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

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.