Olympiad problems

2004 Estonia National Olympiad, 1

Tags: algebra , equation

Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

1980 IMO Shortlist, 12

Tags: diophantine equation , polynomial , equation , number theory , algebra

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

1966 IMO Shortlist, 10

Tags: equation , trigonometric equation , algebra , trigonometry

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

1969 IMO Shortlist, 37

Tags: trigonometric equation , equation , algebra , trigonometry

$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

1971 IMO Shortlist, 8

Tags: root , polynomial , equation , algebra

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

2022 Cyprus JBMO TST, 1

Tags: equation , floor function , algebra

Determine all real numbers $x\in\mathbb{R}$ for which \[ \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022. \] The notation $\lfloor z \rfloor$, for $z\in\mathbb{R}$, denotes the largest integer which is less than or equal to $z$. For example: \[\lfloor 3.98 \rfloor =3 \quad \text{and} \quad \lfloor 0.14 \rfloor =0.\]

1989 IMO Shortlist, 4

Tags: polynomial , diophantine equation , rational roots , equation , algebra

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

2023 Moldova EGMO TST, 7

Tags: equation

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

2006 Estonia Math Open Junior Contests, 1

Tags: equation , algebra

The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$. Find all numbers $n$ such that the value of this expression is $13$.

2013 Bogdan Stan, 4

Tags: algebra , logarithm , equation

Solve in the real numbers the equation $ 3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1. $ [i]Ion Gușatu[/i]

MathLinks Contest 6th, 2.1

Tags: algebra , equation

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

2008 IMO Shortlist, 1

Tags: equation , number theory , algebra

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2016 India National Olympiad, P5

Tags: equation , trigonometry , inradius , geometry

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

2017 District Olympiad, 2

Tags: equation , parameterization , diophantine

Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $ [b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $ [b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $

1996 Greece Junior Math Olympiad, 1

Tags: equation , algebra

Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$

1961 IMO Shortlist, 3

Tags: equation , trigonometric equation , algebra , trigonometry

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

1966 IMO Shortlist, 25

Tags: trigonometric identities , equation , trigonometry , algebra

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

2017 German National Olympiad, 1

Tags: quadratic equation , quadratic , equation , system of equations , algebra

Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.

2017 Balkan MO Shortlist, N1

Tags: algebra , equation

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

2004 Nicolae Coculescu, 2

Tags: logarithm , quadratic , equation , monotone , trigonometry , algebra

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

1966 IMO Shortlist, 40

Tags: equation , parametric equation , algebra

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

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 $a$?

1997 IMO, 5

Tags: prime factorization , equation , number theory

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2004 Nicolae Coculescu, 3

Tags: equation , matrices , cayley-hamilton , linear algebra

Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

1993 Moldova Team Selection Test, 4

Tags: equation

Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$.