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

1988 IMO Longlists, 42
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

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

1960 AMC 12/AHSME, 11
Tags: calculus , integration , vieta
For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

1961 AMC 12/AHSME, 29
Tags: quadratic , algebra , polynomial , 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} $

1996 Taiwan National Olympiad, 4
Tags: polynomial , vieta , algebra
Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.

1958 AMC 12/AHSME, 41
Tags: algebra , polynomial , vieta
The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of \[ x^2 \plus{} px \plus{} q \equal{} 0 \] to be $ r^2$ and $ s^2$, $ p$ must equal: $ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad \textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad \textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\ \textbf{(D)}\ B^2 \minus{} 2C\qquad \textbf{(E)}\ 2C \minus{} B^2$

2014 AIME Problems, 9
Tags: algebra , polynomial , vieta , quadratic , trigonometry
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2000 Harvard-MIT Mathematics Tournament, 6
Tags: algebra , polynomial , vieta
If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

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

2024 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , vieta
Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.

2002 AMC 10, 14
Tags: quadratic , vieta , algebra , number theory , 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}$

2013 Iran Team Selection Test, 5
Tags: modular arithmetic , algebra , polynomial , quadratic , vieta , number theory
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

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 $EF$.

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

2013 AIME Problems, 12
Tags: algebra , polynomial , trigonometry , vieta , complex number
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

2002 Italy TST, 3
Tags: polynomial , vieta , relatively prime , number theory , algebra
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

1979 IMO Shortlist, 3
Tags: algebra , polynomial , vieta , functional equation
Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

2005 International Zhautykov Olympiad, 3
Tags: vieta , number theory
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$

1987 IMO Longlists, 24
Tags: polynomial , vieta , algebra
Prove that if the equation $x^4 + ax^3 + bx + c = 0$ has all its roots real, then $ab \leq 0.$

1989 Federal Competition For Advanced Students, P2, 5
Tags: vieta , algebra
Find all real solutions of the system: $ x^2\plus{}2yz\equal{}x,$ $ y^2\plus{}2zx\equal{}y,$ $ z^2\plus{}2xy\equal{}z.$

1983 AIME Problems, 3
Tags: polynomial , vieta , function , domain , quadratic , algebra
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

2011 All-Russian Olympiad, 2
Tags: quadratic , invariant , vieta , induction , algebra , polynomial , arithmetic sequence
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

2009 USA Team Selection Test, 5
Tags: quadratic , inequalities , algebra , polynomial , vieta , number theory
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]

2005 AIME Problems, 8
Tags: vieta , algebra , polynomial , number theory , logarithm
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

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 $\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$