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

PEN A Problems, 3
Tags: algebra , polynomial , vieta , function , induction , divisibility theory
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.

2019 BMT Spring, 5
Tags: vieta
Find the sum of all real solutions to $ (x^2 - 10x - 12)^{x^2+5x+2} = 1 $

1954 AMC 12/AHSME, 25
Tags: vieta
The two roots of the equation $ a(b\minus{}c)x^2\plus{}b(c\minus{}a)x\plus{}c(a\minus{}b)\equal{}0$ are $ 1$ and: $ \textbf{(A)}\ \frac{b(c\minus{}a)}{a(b\minus{}c)} \qquad \textbf{(B)}\ \frac{a(b\minus{}c)}{c(a\minus{}b)} \qquad \textbf{(C)}\ \frac{a(b\minus{}c)}{b(c\minus{}a)} \qquad \textbf{(D)}\ \frac{c(a\minus{}b)}{a(b\minus{}c)} \qquad \textbf{(E)}\ \frac{c(a\minus{}b)}{b(c\minus{}a)}$

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

2012 ELMO Shortlist, 6
Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory
Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2019 CMIMC, 6
Tags: algebra , polynomial , vieta
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of $$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$

2013 Korea National Olympiad, 7
Tags: vieta , algebra
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows: \[ b_1 = c_1 = 1 \] \[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \] \[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \] Let $a_k = b_{2014} $. Find the value of \[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]

1991 AMC 12/AHSME, 20
Tags: algebra , polynomial , vieta , logarithm , geometry
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 $

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 $

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

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]

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 $P(x)$. $\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$

1954 AMC 12/AHSME, 41
Tags: algebra , polynomial , vieta
The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is: $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \minus{}8 \qquad \textbf{(D)}\ \minus{}2 \qquad \textbf{(E)}\ 0$

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

2011 NIMO Summer Contest, 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 Federal Competition For Advanced Students, Part 2, 2
Tags: algebra , polynomial , vieta , number theory
(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$?

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]

1985 IberoAmerican, 1
Tags: polynomial , vieta , modular arithmetic , symmetry , algebra
Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

PEN A Problems, 82
Tags: quadratic , function , inequalities , vieta , divisibility theory , vieta jumping
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

MathLinks Contest 7th, 7.1
Tags: modular arithmetic , quadratic , vieta , number theory , relatively prime , algebra
Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

2014 Contests, 3
Tags: algebra , polynomial , vieta
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

2010 Tuymaada Olympiad, 3
Tags: quadratic , polynomial , vieta , absolute value , algebra
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

2006 Flanders Math Olympiad, 1
Tags: trigonometry , polynomial , vieta , complex number , algebra
(a) Solve for $\theta\in\mathbb{R}$: $\cos(4\theta) = \cos(3\theta)$ (b) $\cos\left(\frac{2\pi}{7}\right)$, $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$.

1995 AIME Problems, 2
Tags: quadratic , modular arithmetic , algebra , polynomial , vieta , logarithm , search
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

2005 India National Olympiad, 3
Tags: quadratic , vieta , algebra
Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that $a)$ $\alpha$ is real and negative; $b)$ the remaining third quadratic equation has non-real roots.