Found problems: 744
2019 Philippine MO, 3
Find all triples $(a, b, c)$ of positive integers such that
$a^2 + b^2 = n\cdot lcm(a, b) + n^2$
$b^2 + c^2 = n \cdot lcm(b, c) + n^2$
$c^2 + a^2 = n \cdot lcm(c, a) + n^2$
for some positive integer $n$.
1967 IMO Longlists, 6
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
1981 Swedish Mathematical Competition, 2
Does
\[\left\{ \begin{array}{l}
x^y = z \\
y^z = x \\
z^x = y \\
\end{array} \right.
\]
have any solutions in positive reals apart from $x = y = z= 1$?
2005 Kazakhstan National Olympiad, 1
Does there exist a solution in real numbers of the system of equations
\[\left\{
\begin{array}{rcl}
(x - y)(z - t)(z - x)(z - t)^2 = A, \\
(y - z)(t - x)(t - y)(x - z)^2 = B,\\
(x - z)(y - t)(z - t)(y - z)^2 = C,\\
\end{array}
\right.\]
when
a) $A=2, B=8, C=6;$
b) $A=2, B=6, C=8.$?
1967 IMO Shortlist, 5
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
2017 Kosovo National Mathematical Olympiad, 2
Solve the system of equations
$x+y+z=\pi$
$\tan x\tan z=2$
$\tan y\tan z=18$
1989 IberoAmerican, 1
Determine all triples of real numbers that satisfy the following system of equations:
\[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]
2002 IMO Shortlist, 5
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by
\[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\]
where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.
1999 Harvard-MIT Mathematics Tournament, 7
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
2019 Saudi Arabia Pre-TST + Training Tests, 1.1
Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$
Find all possible values of $T = x + y + z$
1972 Bulgaria National Olympiad, Problem 2
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]
2009 JBMO Shortlist, 3
Find all values of the real parameter $a$, for which the system
$(|x| + |y| - 2)^2 = 1$
$y = ax + 5$
has exactly three solutions
1990 IMO Longlists, 24
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:
\[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
2006 Hanoi Open Mathematics Competitions, 3
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations
$x^2 + y -z = 100$ and $x + y^2 - z = 124$:
2015 India Regional MathematicaI Olympiad, 3
Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).
1976 IMO Longlists, 25
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
1976 Vietnam National Olympiad, 1
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
1997 Israel National Olympiad, 1
Find all real solutions to the system of equations
$$\begin{cases} x^2 +y^2 = 6z \\
y^2 +z^2 = 6x \\
z^2 +x^2 = 6y \end{cases}$$
2025 Ukraine National Mathematical Olympiad, 9.1
Solve the system of equations in reals:
\[
\begin{cases}
y = x^2 + 2x \\
z = y^2 + 2y \\
x = z^2 + 2z
\end{cases}
\]
[i]Proposed by Mykhailo Shtandenko[/i]
2014 Finnish National High School Mathematics, 1
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
2018 IMO Shortlist, A2
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
2011 Belarus Team Selection Test, 4
Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$)
a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$
b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)?
D. Bazylev
2016 Costa Rica - Final Round, A1
Find all solutions of the system
$\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $
$\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $
$\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$
$x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$
where $x, y, z \ge 0$ and $w \in R$
[hide=PS] I attached the system, in case I have any typos[/hide]
1980 AMC 12/AHSME, 29
How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below?
\[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \]
$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$
2022 New Zealand MO, 3
Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$
$$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$