Olympiad problems

2007 AMC 12/AHSME, 21

Tags: quadratics , function , Vieta , algebra , quadratic formula

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following? $ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$ $ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$ $ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$ $ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$

2005 India National Olympiad, 3

Tags: quadratics , Vieta , algebra , algebra unsolved

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.

2007 Harvard-MIT Mathematics Tournament, 20

Tags: algebra , polynomial , Vieta

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

2014 AMC 12/AHSME, 19

Tags: AMC , AIME , symmetry , algebra , polynomial , Vieta , AIME I

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$? $\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

2021 Science ON all problems, 3

Tags: complex numbers , 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]

1979 IMO Shortlist, 3

Tags: algebra , polynomial , Vieta , functional equation , IMO Longlist

Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

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)}$

2006 Harvard-MIT Mathematics Tournament, 6

Tags: Vieta

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$.

2004 Indonesia MO, 2

Tags: quadratics , Vieta , algebra

Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.

2012 Iran Team Selection Test, 1

Tags: algebra , polynomial , modular arithmetic , induction , Vieta , calculus , Iran

Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder. [i]Proposed by Yahya Motevassel[/i]

1959 AMC 12/AHSME, 34

Tags: Vieta , irrational number , complex numbers , imaginary numbers , AMC , Quadratic , AMC 12

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}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

1971 IMO Longlists, 16

Tags: algebra , system of equations , Vieta , Polynomials , IMO Shortlist

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

2010 AMC 12/AHSME, 21

Tags: algebra , polynomial , Vieta , analytic geometry , AMC

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$

1965 AMC 12/AHSME, 7

Tags: algebra , polynomial , Vieta

The sum of the reciprocals of the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ is: $ \textbf{(A)}\ \frac {1}{a} \plus{} \frac {1}{b} \qquad \textbf{(B)}\ \minus{} \frac {c}{b} \qquad \textbf{(C)}\ \frac {b}{c} \qquad \textbf{(D)}\ \minus{} \frac {a}{b} \qquad \textbf{(E)}\ \minus{} \frac {b}{c}$

2010 AMC 10, 19

Tags: geometry , Vieta , ratio , AMC , AMC 10 , trig identities

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$

PEN A Problems, 6

Tags: algebra , polynomial , Vieta , Divisibility Theory , pen

[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

2014 India Regional Mathematical Olympiad, 4

Tags: number theory , least common multiple , inequalities , symmetry , Vieta , algebra , system of equations

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

2007 AMC 12/AHSME, 14

Tags: Vieta

Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be distinct integers such that \[ (6 \minus{} a)(6 \minus{} b)(6 \minus{} c)(6 \minus{} d)(6 \minus{} e) \equal{} 45. \]What is $ a \plus{} b \plus{} c \plus{} d \plus{} e?$ $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

2010 Tuymaada Olympiad, 3

Tags: quadratics , algebra , polynomial , Vieta , absolute value , algebra unsolved

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 $f(1)=0$.

2019 CMIMC, 6

Tags: algebra , polynomial , Vieta , 2019

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

2010 AMC 12/AHSME, 23

Tags: quadratics , algebra , polynomial , symmetry , Vieta , AMC

Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$? $ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$

2005 AIME Problems, 13

Tags: algebra , polynomial , quadratics , Vieta , function , AMC , AIME

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 $n_1\cdot n_2$.

1985 Traian Lălescu, 1.1

Tags: algebra , polynomial , Vieta

$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations $$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$ Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.

1988 IMO Longlists, 27

Tags: trigonometry , algebra , polynomial , Vieta , function , algebra unsolved

Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

2001 National Olympiad First Round, 11

Tags: quadratics , Vieta , SFFT

For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7\\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of the preceding} $