Found problems: 451
2010 VTRMC, Problem 3
Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$
2003 Gheorghe Vranceanu, 4
Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $
1996 Akdeniz University MO, 1
Solve the equation for real numbers $x,y,z$
$$(x-y+z)^2=x^2-y^2+z^2$$
2000 Moldova National Olympiad, Problem 5
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
2016 Kazakhstan National Olympiad, 2
Find all rational numbers $a$,for which there exist infinitely many positive rational numbers $q$ such that the equation
$[x^a].{x^a}=q$ has no solution in rational numbers.(A.Vasiliev)
1970 IMO Shortlist, 5
Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that
\[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\]
($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).
1980 IMO Longlists, 12
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
1995 IMO Shortlist, 2
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
\[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]
2004 Nicolae Coculescu, 1
Find all pairs of integers $ (a,b) $ such that the equation
$$ |x-1|+|x-a|+|x-b|=1 $$
has exactly one real solution.
[i]Florian Dumitrel[/i]
2018 Ramnicean Hope, 2
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $
[i]Ovidiu Țâțan[/i]
2008 Hanoi Open Mathematics Competitions, 4
Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation
\[x^3-nx+mn=0\]
has three distint integer roots.
2017 EGMO, 5
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$
2014 Contests, 1
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
2016 Kosovo Team Selection Test, 1
Solve equation in real numbers
$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$
2007 Mathematics for Its Sake, 3
Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $
[i]Dumitru Acu[/i]
1997 Romania National Olympiad, 4
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$
If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.
2011 Dutch BxMO TST, 3
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
1982 IMO Longlists, 31
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
2003 Poland - Second Round, 6
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.
2011 IMO Shortlist, 2
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]
1989 IMO Longlists, 4
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
2016 District Olympiad, 1
Solve in $ \mathbb{N}^2: $
$$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$
2015 Caucasus Mathematical Olympiad, 1
Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given).
2017 China Team Selection Test, 1
Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$