Olympiad problems

1937 Moscow Mathematical Olympiad, 032

Solve the system $\begin{cases} x+ y +z = a \\ x^2 + y^2 + z^2 = a^2 \\ x^3 + y^3 +z^3 = a^3 \end{cases}$

1939 Moscow Mathematical Olympiad, 043

Tags: algebra , parameter , system of equations

Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\ x + y+ z = 2b \\ x^2 + y^2-z^2 = b^2 \end{cases}$ in $C$

1990 Czech and Slovak Olympiad III A, 2

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

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.

1978 Chisinau City MO, 161

Tags: equation , parameter , algebra

For what real values of $a$ the equation $\frac{2^{2x}}{2^{2x}+2^{x+1}+1}+a \frac{2^x}{2^x+1}+(a-1) = 0$ has a single root ?

1935 Moscow Mathematical Olympiad, 017

Tags: algebra , parameter , system of equations

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]

IV Soros Olympiad 1997 - 98 (Russia), 11.12

Tags: algebra , system of equations , parameter

Find how many different solutions depending on $a$ has the system of equations : $$\begin{cases} x+z=2a \\ y+u+xz=a-3 \\ yz+xu=2a \\ yu=1 \end{cases}$$

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , parameter , system of equations

Solve the system $\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}$

2024 Austrian MO National Competition, 1

Tags: algebra , parameter , parameterization , functional equation

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]

1996 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , parameter , inequalities

$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$ $b)$ Does inequality $a)$ holds for $1)$ arbitrary real numbers $a$, $b$ and $c$ $2)$ any integer $m$

2007 Bulgarian Autumn Math Competition, Problem 8.1

Tags: algebra , system of equations , parameter , parameterization

Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

1990 Czech and Slovak Olympiad III A, 4

Tags: max , inequalities , algebra , parameter

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

1982 Polish MO Finals, 3

Tags: algebra , system of equations , parameter

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.

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: inequalities , algebra , parameter

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