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.

AND:
OR:
NO:

Found problems: 236

1969 AMC 12/AHSME, 5
Tags: Vieta , AMC
If a number $N$, $N\neq 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is: $\textbf{(A) }\dfrac1R\qquad \textbf{(B) }R\qquad \textbf{(C) }4\qquad \textbf{(D) }\dfrac14\qquad \textbf{(E) }-R$

2010 Bulgaria National Olympiad, 2
Tags: geometry , function , rhombus , Vieta , geometry proposed
Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

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$

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

2011 AIME Problems, 15
Tags: algebra , polynomial , Vieta , quadratics , AMC , AIME , YUH.
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.

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

2013 Turkey Junior National Olympiad, 1
Tags: inequalities , algebra , polynomial , Vieta , analytic geometry , function , calculus
Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

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]

2009 Today's Calculation Of Integral, 417
Tags: calculus , integration , function , geometry , algebra , polynomial , Vieta
The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

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

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$

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$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

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

1966 AMC 12/AHSME, 30
Tags: Vieta , AMC
If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is: $\text{(A)}\ 35 \qquad \text{(B)}\ 24\qquad \text{(C)}\ -12\qquad \text{(D)}\ -61 \qquad \text{(E)}\ -63$

1998 Harvard-MIT Mathematics Tournament, 7
Tags: algebra , polynomial , Vieta
Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$?

1982 Vietnam National Olympiad, 1
Tags: algebra , polynomial , calculus , integration , trigonometry , Vieta , algebra proposed
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$

2012 India Regional Mathematical Olympiad, 2
Tags: algebra , polynomial , Vieta , inequalities , binomial theorem , sum of roots
Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

1977 AMC 12/AHSME, 23
Tags: quadratics , geometry , 3D geometry , Vieta , AMC
If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then $\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$ $\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$

1988 IMO Shortlist, 16
Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988
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.

PEN A Problems, 82
Tags: quadratics , 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?

1981 IMO, 3
Tags: algebra , polynomial , Vieta , number theory , IMO , IMO 1981 , Diophantine equation
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2021 India National Olympiad, 2
Tags: algebra , Cubics , integer roots , Vieta , INMO , v_2
Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

2002 Italy TST, 3
Tags: algebra , polynomial , Vieta , number theory , relatively prime , number theory unsolved
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 $x^2+m.$

2004 Romania Team Selection Test, 6
Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

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