Olympiad problems

2003 Switzerland Team Selection Test, 1

Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$ Find all possible values of $a$.

IV Soros Olympiad 1997 - 98 (Russia), 11.12

Tags: system of equations , algebra , parameter

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

1939 Moscow Mathematical Olympiad, 046

Tags: radical , parameter , equation , algebra

Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.

1945 Moscow Mathematical Olympiad, 097

Tags: system of equations , parameter , algebra

The system $\begin{cases} x^2 - y^2 = 0 \\ (x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?

1996 Bosnia and Herzegovina Team Selection Test, 1

Tags: inequalities , algebra , parameter

$a)$ 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 $m$

1999 Estonia National Olympiad, 2

Tags: algebra , parameter , trinomial

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.

2007 Bulgarian Autumn Math Competition, Problem 12.1

Tags: trigonometric equation , parameter , parameterization , algebra

Determine the values of the real parameter $a$, such that the equation \[\sin 2x\sin 4x-\sin x\sin 3x=a\] has a unique solution in the interval $[0,\pi)$.

2022 Bulgaria National Olympiad, 4

Tags: algebra , system of equations , parameter

Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. If there exists an $a>0$ such that \[x_{i}^2=a+x_{i+1}x_{i+2}\quad\forall 1\leq i\leq n\] then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , system of equations , parameter

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

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 $\frac{32}{225}$.

1976 Dutch Mathematical Olympiad, 4

Tags: algebra , parameter , parameterization , equation , trinomial

For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?

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.

1935 Moscow Mathematical Olympiad, 010

Tags: system of equations , algebra , parameter

Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\ x + y + 2z = 4(a^2 + 1) \\ z^2 - xy = a^2 \end{cases}$

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 ?

1965 German National Olympiad, 1

Tags: parameter , equation , algebra

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

1954 Czech and Slovak Olympiad III A, 1

Tags: quadratic polynomial , equation , parameter , parameterization

Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.

1983 Spain Mathematical Olympiad, 4

Tags: algebra , equation , parameter

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

1974 Chisinau City MO, 74

Tags: algebra , cubic , parameter

Solve the equation: $x^3-2ax^2+(a^2-2\sqrt2 a -6)x + 2\sqrt2 a^2+ 8a + 4\sqrt2 =0$

1949 Moscow Mathematical Olympiad, 161

Tags: radical , equation , algebra , parameter

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

1990 Czech and Slovak Olympiad III A, 4

Tags: inequalities , algebra , max , 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 $x_{n+1}=x_1$)

1937 Moscow Mathematical Olympiad, 032

Tags: algebra , system of equations , parameter

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 , system of equations , parameter

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 $C$

2000 German National Olympiad, 1

Tags: system of equations , parameter , algebra

For each real parameter $a$, find the number of real solutions to the system $$\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}$$

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.

2002 Swedish Mathematical Competition, 3

Tags: parabola , system of equations , system , parameter , algebra , analytic geometry

$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?