Olympiad problems

1997 Finnish National High School Mathematics Competition, 1

Tags: quadratic , product of roots , parameterization , logarithm , polynomial , algebra

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

MathLinks Contest 7th, 5.1

Tags: product of roots , inequalities , vieta , algebra , absolute value , polynomial

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

1985 AMC 12/AHSME, 30

Tags: algebra , polynomial , product of roots , sum of roots , floor function

Let $ \lfloor x \rfloor$ be the greatest integer less than or equal to $ x$. Then the number of real solutions to $ 4x^2 \minus{} 40 \lfloor x \rfloor \plus{} 51 \equal{} 0$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2011 Purple Comet Problems, 9

Tags: polynomial , quadratic , quadratic formula , sum of roots , product of roots , algebra

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

2001 AMC 12/AHSME, 19

Tags: polynomial , product of roots , algebra

The polynomial $ P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $ y$-intercept of the graph of $ y \equal{} P(x)$ is 2, what is $ b$? $ \textbf{(A)} \ \minus{} 11 \qquad \textbf{(B)} \ \minus{} 10 \qquad \textbf{(C)} \ \minus{} 9 \qquad \textbf{(D)} \ 1 \qquad \textbf{(E)} \ 5$

2007 Belarusian National Olympiad, 6

Tags: polynomial , algebra , product of roots

Let $a$ be the sum and $b$ the product of the real roots of the equation $x^4-x^3-1=0$ Prove that $b < -\frac{11}{10}$ and $a > \frac{6}{11}$.

1996 Canadian Open Math Challenge, 1

Tags: polynomial , algebra , quadratic , quadratic formula , sum of roots , product of roots , rational root theorem

The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?