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

2020/2021 Tournament of Towns, P2
Tags: geometry , algebra , polynomial , vieta
Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?

2002 AMC 12/AHSME, 12
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}$

2004 AMC 12/AHSME, 17
Tags: vieta , logarithm
For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

2013 Stanford Mathematics Tournament, 23
Tags: vieta
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.

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

1979 IMO Longlists, 10
Tags: algebra , polynomial , vieta , functional equation
Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

2021 Science ON grade X, 3
Tags: complex number , newton sums , vieta , algebra
Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

PEN B Problems, 5
Tags: modular arithmetic , algebra , polynomial , vieta , symmetry , search , induction
Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

2005 AIME Problems, 13
Tags: polynomial , quadratic , vieta , function , algebra
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.

2010 Harvard-MIT Mathematics Tournament, 3
Tags: calculus , algebra , polynomial , vieta
Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeroes of $p^\prime (x)$ are also zeroes of $p(x)$. Find $p$. Note: monic means that the leading coefficient is $1$.

2007 Bosnia Herzegovina Team Selection Test, 4
Tags: polynomial , inequalities , vieta , algebra
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?