Found problems: 744
2023 Brazil National Olympiad, 3
Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.
1982 Polish MO Finals, 3
Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations
$$\begin{cases} x^2 +y^2 = a^2 +b^2 \\
x^3 +y^3 = a^3 +b^3 \end{cases}$$
where $a$ and $b$ are given positive numbers.
1965 All Russian Mathematical Olympiad, 063
Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.
1935 Moscow Mathematical Olympiad, 010
Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\
x + y + 2z = 4(a^2 + 1) \\
z^2 - xy = a^2
\end{cases}$
1991 Hungary-Israel Binational, 4
Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.
2002 District Olympiad, 2
Solve in $ \mathbb{C}^3 $ the following chain of equalities:
$$ x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3. $$
1999 Austrian-Polish Competition, 3
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$
$$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$
2008 JBMO Shortlist, 5
Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$
PEN C Problems, 2
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
2015 Junior Balkan Team Selection Tests - Romania, 2
Find all the triplets of real numbers $(x , y , z)$ such that :
$y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$
1979 Swedish Mathematical Competition, 1
Solve the equations:
\[\left\{ \begin{array}{l}
x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\
2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\
3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\
\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\
(n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\
n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1
\end{array} \right.
\]
2013 Dutch IMO TST, 1
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
2017 German National Olympiad, 1
Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$:
\begin{align*} x^2+py+q&=0,\\
y^2+px+q&=0.
\end{align*}
Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.
2024 Polish MO Finals, 4
Do there exist real numbers $a,b,c$ such that the system of equations
\begin{align*}
x+y+z&=a\\
x^2+y^2+z^2&=b\\
x^4+y^4+z^4&=c
\end{align*}
has infinitely many real solutions $(x,y,z)$?
1999 Switzerland Team Selection Test, 8
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
2022 BMT, 17
Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations
$$a^3 - 2a = abc - b - c$$
$$b^3 - 2b = bca - c - a$$
$$c^3 - 2c = cab - a - b.$$
1976 Chisinau City MO, 124
Find $3$ numbers, each of which is equal to the square of the difference of the other two.
2021 Saint Petersburg Mathematical Olympiad, 1
Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$ $$\sin^2{y} + \cos^2{x} = x^2. $$
[i]A. Khrabov[/i]
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
2010 Contests, 2
Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.
1967 IMO Longlists, 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}
$
2016 Rioplatense Mathematical Olympiad, Level 3, 2
Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
2021 Dutch BxMO TST, 2
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
1968 IMO Shortlist, 4
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
2016 Korea Summer Program Practice Test, 1
Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.)
\[ x_i^2 + x_i - 1 = x_{i+1} \]