Olympiad problems

1986 Traian Lălescu, 2.1

Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.

2013 Greece Team Selection Test, 2

Tags: parametric equation , analytic geometry , parameterization

For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

1982 IMO Longlists, 14

Tags: parametric equation , diophantine equation , geometric progression , polynomial , algebra

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

1966 IMO Longlists, 40

Tags: parametric equation , algebra , equation

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

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)

2013 Greece Team Selection Test, 2

Tags: parametric equation , parameterization , analytic geometry

For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

1966 IMO Shortlist, 40

Tags: equation , parametric equation , algebra

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

1966 IMO Shortlist, 48

Tags: number theory , root , parametric equation , equation , algebra

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

1967 IMO Shortlist, 2

Tags: algebra , diophantine equation , parametric equation , root , polynomial

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

1967 IMO Longlists, 43

Tags: polynomial , diophantine equation , parametric equation , root , algebra

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

2016 Indonesia TST, 1

Tags: algebra , parametric equation

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

1979 IMO Shortlist, 6

Tags: parametric equation

Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).

1966 IMO Longlists, 48

Tags: equation , root , parametric equation , number theory , algebra

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

1982 IMO Shortlist, 4

Tags: algebra , polynomial , diophantine equation , parametric equation , geometric progression

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2017 Hanoi Open Mathematics Competitions, 6

Tags: number theory , diophantine , parametric equation , system of equations , diophantine equation

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

1963 IMO Shortlist, 1

Tags: algebra , equation , parametric equation

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

2016 Indonesia TST, 1

Tags: algebra , parametric equation

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

1964 Vietnam National Olympiad, 2

Tags: algebra , parametric equation , quadratic

Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.

1963 IMO, 1

Tags: algebra , equation , parametric equation

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

1957 Czech and Slovak Olympiad III A, 1

Tags: equation , parameterization , algebra , parametric equation

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

2022 Bulgarian Autumn Math Competition, Problem 9.1

Tags: polynomial , algebra , parametric equation , vieta

Given is the equation: \[x^2+mx+2022=0\] a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers

1979 IMO Longlists, 17

Tags: parametric equation

Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).