Olympiad problems

1949-56 Chisinau City MO, 17

Prove that if the roots of the equation $x^2 + px + q = 0$ are real, then for any real number $a$ the roots of the equation $$x^2 + px + q + (x + a) (2x + p) = 0$$ are also real.

1935 Moscow Mathematical Olympiad, 017

Tags: system of equations , algebra , parameter

Solve the system $\begin{cases} x^3 - y^3 = 26 \\ x^2y - xy^2 = 6 \end{cases}$ in $C$ [hide=other version]solved below Solve the system $\begin{cases} x^3 - y^3 = 2b \\ x^2y - xy^2 = b \end{cases}$[/hide]

1965 Czech and Slovak Olympiad III A, 3

Tags: algebra , parameter , radical , equation , real root

Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.

1983 Austrian-Polish Competition, 5

Tags: parameter , system of equations , algebra

Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system $$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\ a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$ has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.

1982 Polish MO Finals, 3

Tags: system of equations , parameter , algebra

Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations $$\begin{cases} x^2 +y^2 = a^2 +b^2 \\ x^3 +y^3 = a^3 +b^3 \end{cases}$$ where $a$ and $b$ are given positive numbers.

1990 Czech and Slovak Olympiad III A, 2

Tags: inequalities , triangle inequality , parameter , positive , quadratic , algebra

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

1999 Estonia National Olympiad, 2

Tags: parameter , trinomial , algebra

Find all values of $a$ such that absolute value of one of the roots of the equation $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$ is twice of absolute value of the other root.

1940 Moscow Mathematical Olympiad, 058

Tags: system of equations , algebra , parameter

Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\ x + y = b \end{cases}$ in $C$

2008 Bulgarian Autumn Math Competition, Problem 8.1

Tags: parameter , algebra , absolute value

Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.

1965 German National Olympiad, 1

Tags: parameter , equation , algebra

For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.

III Soros Olympiad 1996 - 97 (Russia), 9.2

Tags: algebra , parameter , radical

How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$

1995 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , polynomial , algebra , parameter

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

1983 Spain Mathematical Olympiad, 4

Tags: algebra , equation , parameter

Determine the number of real roots of the equation $$16x^5 - 20x^3 + 5x + m = 0.$$