Found problems: 744
2021 Dutch IMO TST, 2
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2004 China Team Selection Test, 1
Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$):
\[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]
2008 All-Russian Olympiad, 5
Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]
1998 Tournament Of Towns, 4
For some positive numbers $A, B, C$ and $D$, the system of equations
$$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$
has $m$ solutions, while the system of equations
$$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$
has $n$ solutions. If $m > n > 1$, find $m$ and $n$.
( G Galperin)
1975 Czech and Slovak Olympiad III A, 3
Determine all real tuples $\left(x_1,x_2,x_3,x_4,x_5,x_6\right)$ such that
\begin{align*}
x_1(x_6 + x_2) &= x_3 + x_5, \\
x_2(x_1 + x_3) &= x_4 + x_6, \\
x_3(x_2 + x_4) &= x_5 + x_1, \\
x_4(x_3 + x_5) &= x_6 + x_2, \\
x_5(x_4 + x_6) &= x_1 + x_3, \\
x_6(x_5 + x_1) &= x_2 + x_4.
\end{align*}
2017 Azerbaijan Junior National Olympiad, P1
Solve the system of equation for $(x,y) \in \mathbb{R}$
$$\left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\
3x^2+4xy=24
\end{matrix}\right.$$ \\
Explain your answer
1987 AMC 12/AHSME, 9
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $
2018 IMO, 2
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]
2017 Finnish National High School Mathematics Comp, 2
Determine $x^2+y^2$ and $x^4+y^4$, when $x^3+y^3=2$ and $x+y=1$
2010 Saudi Arabia BMO TST, 4
Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.
1957 Moscow Mathematical Olympiad, 366
Solve the system: $$\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}$$
1978 IMO Longlists, 53
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.
2015 Kosovo Team Selection Test, 3
It's given system of equations
$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$
..........
$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$
such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
2024 Francophone Mathematical Olympiad, 1
Find the largest integer $k$ with the following property: Whenever real numbers $x_1,x_2,\dots,x_{2024}$ satisfy
\[x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,\]
at least $k$ of them are equal.
1994 Tournament Of Towns, (433) 3
Let $a, b, c$ and $d$ be real numbers such that
$$a^3+b^3+c^3+d^3=a+b+c+d=0$$
Prove that the sum of a pair of these numbers is equal to $0$.
(LD Kurliandchik)
1999 Junior Balkan Team Selection Tests - Moldova, 1
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
2012 Vietnam National Olympiad, 2
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
2013 NIMO Problems, 5
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
1964 Putnam, A2
Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that
$$\int_{0}^{1} f(x)\; dx =1, $$
$$\int_{0}^{1} xf(x)\; dx =\alpha,$$
$$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$
where $\alpha$ is a given real number.
2016 Japan MO Preliminary, 7
Let $a, b, c, d$ be real numbers satisfying the system of equation
$\[(a+b)(c+d)=2 \\
(a+c)(b+d)=3 \\
(a+d)(b+c)=4\]$
Find the minimum value of $a^2+b^2+c^2+d^2$.
2018 Canadian Mathematical Olympiad Qualification, 1
Determine all real solutions to the following system of equations:
$$
\begin{cases}
y = 4x^3 + 12x^2 + 12x + 3\\
x = 4y^3 + 12y^2 + 12y + 3.
\end{cases}
$$
III Soros Olympiad 1996 - 97 (Russia), 10.4
Solve the system of equations
$$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$
2011 Middle European Mathematical Olympiad, 8
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
2015 Abels Math Contest (Norwegian MO) Final, 1a
Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\
y^2 + 4z^2 = 4xy \\
z^2 + 4x^2 = 4yz \end{cases}$
III Soros Olympiad 1996 - 97 (Russia), 9.7
Solve the system of equations:
$$\begin{cases} xy+zu=14
\\ xz+yu=11
\\ xu+yz=10
\\ x+y+z+u=10
\end{cases}$$