Olympiad problems

1940 Moscow Mathematical Olympiad, 058

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

1983 Spain Mathematical Olympiad, 4

Tags: parameter , equation , algebra

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

2022 Bulgaria National Olympiad, 4

Tags: parameter , system of equations , algebra

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.

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.

1995 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , parameter , algebra , polynomial

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.

2020 Czech and Slovak Olympiad III A, 3

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

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)

2007 Bulgarian Autumn Math Competition, Problem 11.2

Tags: parameter , inequalities , 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.

1976 Dutch Mathematical Olympiad, 4

Tags: parameter , equation , parameterization , trinomial , algebra

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 $a$?

1974 Chisinau City MO, 74

Tags: parameter , cubic , algebra

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

1965 German National Olympiad, 1

Tags: parameter , equation , algebra

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

1949-56 Chisinau City MO, 17

Tags: parameter , trinomial , algebra

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.

2000 German National Olympiad, 1

Tags: parameter , algebra , system of equations

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

2008 Bulgarian Autumn Math Competition, Problem 8.1

Tags: parameter , absolute value , algebra

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

2008 Bulgarian Autumn Math Competition, Problem 10.1

Tags: parameter , real root , 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.

1965 Czech and Slovak Olympiad III A, 3

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

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: algebra , parameter , system of equations

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.

1940 Moscow Mathematical Olympiad, 058

1983 Spain Mathematical Olympiad, 4

2022 Bulgaria National Olympiad, 4

1999 Estonia National Olympiad, 2

1995 Czech And Slovak Olympiad IIIA, 6

2020 Czech and Slovak Olympiad III A, 3

2007 Bulgarian Autumn Math Competition, Problem 11.2

1976 Dutch Mathematical Olympiad, 4

1974 Chisinau City MO, 74

1965 German National Olympiad, 1

1949-56 Chisinau City MO, 17

2000 German National Olympiad, 1

2008 Bulgarian Autumn Math Competition, Problem 8.1

2008 Bulgarian Autumn Math Competition, Problem 10.1

1965 Czech and Slovak Olympiad III A, 3

1983 Austrian-Polish Competition, 5

2002 Swedish Mathematical Competition, 3

2003 Switzerland Team Selection Test, 1

1945 Moscow Mathematical Olympiad, 097

1939 Moscow Mathematical Olympiad, 046

1935 Moscow Mathematical Olympiad, 010

2007 Bulgarian Autumn Math Competition, Problem 12.1

III Soros Olympiad 1996 - 97 (Russia), 9.2

1954 Czech and Slovak Olympiad III A, 1

1949 Moscow Mathematical Olympiad, 161