Found problems: 451
2001 Estonia National Olympiad, 4
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.
2019 Irish Math Olympiad, 4
Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy
$$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$
2000 Romania National Olympiad, 4
Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $
1986 IMO Longlists, 18
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
2004 Gheorghe Vranceanu, 4
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $
2011 India IMO Training Camp, 2
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]
1978 Chisinau City MO, 161
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 ?
2023 Poland - Second Round, 3
Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
2021 Hong Kong TST, 1
Find all real triples $(a,b,c)$ satisfying
\[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]
1978 USAMO, 3
An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.
2012 Centers of Excellency of Suceava, 4
Solve in the reals the following system.
$$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$
[i]Gheorghe Marchitan[/i]
2007 Nicolae Coculescu, 2
[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $
[b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $
[i]Florian Dumitrel[/i]
1989 ITAMO, 1
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.
2000 Tournament Of Towns, 1
Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$
(RM Kuznec)
1966 IMO Longlists, 48
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
1971 IMO Longlists, 31
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 $x_3, x_1.$
2009 ISI B.Math Entrance Exam, 2
Let $c$ be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.
1967 IMO Longlists, 44
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
2019 Finnish National High School Mathematics Comp, 1
Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$
1954 Moscow Mathematical Olympiad, 272
Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.
1969 Czech and Slovak Olympiad III A, 1
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]
2009 Kyiv Mathematical Festival, 1
Solve the equation $\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x$, $x\in \big[0,\frac{\pi}{2}\big)$
Taiwan TST 2015 Round 1, 1
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
2016 District Olympiad, 2
If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $
1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.
1978 Vietnam National Olympiad, 1
Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.