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

2010 Math Prize For Girls Problems, 20
Tags: polynomial , function , vieta , complex number , trigonometry , algebra , ratio
What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

2013 NIMO Problems, 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]

2010 AMC 12/AHSME, 17
Tags: trig identities , geometry , vieta , ratio
Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$? $ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad \textbf{(B)}\ \frac {10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac {17}{4} \qquad \textbf{(E)}\ 6$

MathLinks Contest 7th, 4.3
Tags: polynomial , parameterization , vieta , cauchy inequality , inequalities , function , algebra
Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

2006 Pan African, 2
Tags: number theory , polynomial , algebra , vieta , rational root theorem
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.

2010 Contests, 1
Tags: functional equation , function , algebra , vieta
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

1998 All-Russian Olympiad, 1
Tags: conic , geometry , polynomial , algebra , parabola , vieta
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.

2010 AMC 10, 9
Tags: algebra , polynomial , vieta
A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

1959 AMC 12/AHSME, 34
Tags: vieta , irrational number , complex number , imaginary numbers , quadratic
Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

2002 AMC 10, 14
Tags: number theory , algebra , quadratic , vieta , prime number , special factorizations
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2008 Harvard-MIT Mathematics Tournament, 6
Tags: trigonometry , polynomial , quadratic , hmmt , vieta , algebra
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.

1998 All-Russian Olympiad, 1
Tags: analytic geometry , vieta , algebra
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2010 AMC 12/AHSME, 21
Tags: algebra , vieta , analytic geometry , polynomial
The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2008 IMO, 2
Tags: inequalities , algebra , calculus , vieta
[b](a)[/b] Prove that \[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [i]Author: Walther Janous, Austria[/i]

1998 All-Russian Olympiad, 1
Tags: algebra , analytic geometry , vieta
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2008 ITest, 53
Tags: polynomial , algebra , vieta
Find the sum of the $2007$ roots of \[(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007).\]

2015 AMC 12/AHSME, 12
Tags: polynomial , vieta , algebra
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 $

2014 AIME Problems, 5
Tags: function , algebra , quadratic , vieta , polynomial
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.

2010 Stanford Mathematics Tournament, 10
Tags: algebra , polynomial , vieta
Find the sum of all solutions of the equation $\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$

2006 Princeton University Math Competition, 3
Tags: algebra , vieta , polynomial
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?

1995 South africa National Olympiad, 2
Tags: algebra , polynomial , number theory , function , vieta
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.

1970 AMC 12/AHSME, 11
Tags: algebra , polynomial , vieta
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is $\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$

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

2020 LIMIT Category 1, 7
Tags: limit , algebra , vieta , polynomial
Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$ (A)$5$ (B)$6$ (C)$7$ (D)$8$

2008 AIME Problems, 7
Tags: algebra , polynomial , vieta
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$.