Olympiad problems

1996 Spain Mathematical Olympiad, 4

For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.

1949-56 Chisinau City MO, 53

Tags: equation , radical , algebra

Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$

2004 Thailand Mathematical Olympiad, 5

Tags: equation , algebra , sum , radical

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$

2013 IFYM, Sozopol, 6

Tags: equation , integer root , algebra

For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?

1994 IMO Shortlist, 5

Tags: equation , number theory , binary representation , function

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

2025 Bangladesh Mathematical Olympiad, P2

Tags: algebra , exponential equation , equation

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

1968 IMO Shortlist, 22

Tags: decimal representation , digit , equation , number theory

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2012 India IMO Training Camp, 1

Tags: equation , sequence , algebra

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

2012 Greece JBMO TST, 1

Tags: equation , system of equations , algebra

Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$

1949-56 Chisinau City MO, 54

Tags: equation , algebra

Solve the equation: $$\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)$$

2005 Grigore Moisil Urziceni, 1

Tags: equation , algebra , system of equations

Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

2011 Ukraine Team Selection Test, 7

Tags: equation , lte lemma , modular arithmetic , number theory

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2006 Grigore Moisil Urziceni, 3

Tags: equation , monotone , algebra , system of equations

Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

2007 Alexandru Myller, 1

Tags: equation , perfect square , diophantine equation , divisibility , number theory

Solve $ x^3-y^3=2xy+7 $ in integers.

1967 IMO Longlists, 48

Tags: equation , root , exponential equation , algebra

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

2016 Junior Regional Olympiad - FBH, 4

Tags: equation , number theory

In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$

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.

2003 Estonia National Olympiad, 2

Tags: equation , floor function , algebra

Find all positive integers $n$ such that $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$

2006 Petru Moroșan-Trident, 2

Tags: system of equations , algebra , monotone , equation

Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

1993 Swedish Mathematical Competition, 4

Tags: equation , algebra

To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$. Solve the equation $x*36 = 216$.

2015 Hanoi Open Mathematics Competitions, 13

Tags: equation , rational , algebra

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

2009 District Olympiad, 3

Tags: logarithm , equation , algebra

Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

1972 Bulgaria National Olympiad, Problem 2

Tags: algebra , equation , system of equations

Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$. [i]H. Lesov[/i]

2015 Hanoi Open Mathematics Competitions, 3

Tags: equation , algebra

Suppose that $a > b > c > 1$. One of solutions of the equation $\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$ is (A): $-1$, (B): $-2$, (C): $0$, (D): $1$, (E): None of the above.

1992 IMO Longlists, 14

Tags: equation , divisibility , modular arithmetic , number theory

Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and \[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\] where $a_{n+j} = a_j$. Prove that $n \neq 1992.$