Olympiad problems

III Soros Olympiad 1996 - 97 (Russia), 9.2

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

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.

2007 Bulgarian Autumn Math Competition, Problem 11.2

Tags: inequalities , parameter , algebra

Find all values of the parameter $a$ for which the inequality \[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\] has a unique solution.

2008 Bulgarian Autumn Math Competition, Problem 10.1

Tags: real root , parameter , algebra

For which values of the parameter $a$ does the equation \[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\] has three different real roots.

1949-56 Chisinau City MO, 17

Tags: algebra , trinomial , parameter

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.

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: algebra , parameter , inequalities

Find all values of the parameter $a$ for which there exist exactly two integer values of $x$ that satisfy the inequality $$x^2+5\sqrt2 x+a<0.$$

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 $m$.

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 $C$

2024 Austrian MO National Competition, 1

Tags: functional equation , parameter , parameterization , algebra

Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

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.

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]

1990 Czech and Slovak Olympiad III A, 2

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

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.

2020 Czech and Slovak Olympiad III A, 3

Tags: parametric equation , system of equations , algebra , parameter

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\ y^2 - 3z + p = x, \\ z^2 - 3x + p = y \end{cases}$ with real parameter $p$. a) For $p \ge 4$, solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$. (Jaroslav Svrcek)