Found problems: 744
2010 Dutch IMO TST, 5
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
2004 Peru MO (ONEM), 3
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
1994 Vietnam National Olympiad, 1
Find all real solutions to
\[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\]
\[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\]
\[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]
1989 Greece National Olympiad, 1
Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix}
ax -by-c z-dw =0\\
b x +a y -d z +cw=0\\
c x+ d y +a z -b w=0\\
dx-c y+bz+aw=0
\end{matrix}$$
prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$
2007 Cuba MO, 1
Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$
2004 Croatia National Olympiad, Problem 1
Find all real solutions of the system of equations
$$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$
2010 Baltic Way, 1
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations
\[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]
2005 Junior Tuymaada Olympiad, 3
Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?
1984 All Soviet Union Mathematical Olympiad, 382
Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\
y^2/ 3 + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$
Find the value of expression $xy + 2yz + 3zx$.
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:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
1971 IMO Longlists, 16
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
1954 Moscow Mathematical Olympiad, 274
Solve the system $\begin{cases}
10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\
11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\
15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\
2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\
6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\
3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\
4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$
1988 IMO Longlists, 62
Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.
1983 IMO Longlists, 54
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
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]
2011 Regional Competition For Advanced Students, 2
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true:
\begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\
\left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}
2019 Saudi Arabia Pre-TST + Training Tests, 3.2
Find all triples of real numbers $(x, y,z)$ such that
$$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$
2019 German National Olympiad, 6
Suppose that real numbers $x,y$ and $z$ satisfy the following equations:
\begin{align*}
x+\frac{y}{z} &=2,\\
y+\frac{z}{x} &=2,\\
z+\frac{x}{y} &=2.
\end{align*}
Show that $s=x+y+z$ must be equal to $3$ or $7$.
[i]Note:[/i] It is not required to show the existence of such numbers $x,y,z$.
1953 Moscow Mathematical Olympiad, 248
a) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}$
b) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\
... \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$
1983 Swedish Mathematical Competition, 3
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
2008 All-Russian Olympiad, 5
Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]
2021 Baltic Way, 5
Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.
2006 MOP Homework, 4
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.
2010 Saint Petersburg Mathematical Olympiad, 1
Solve in positives $$x^y=z,y^z=x,z^x=y$$
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$