Found problems: 744
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations:
$a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
The five real numbers $v,w,x,y,s$ satisfy the system of equations
\begin{align*}
v&=wx+ys,\\
v^2&=w^2x+y^2s,\\
v^3&=w^3x+y^3s.
\end{align*}
Show that at least two of them are equal.
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations
$$a^2+ab=c$$
$$b^2+bc=a$$
$$c^2+ca=b$$
Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously:
\begin{align*}
x^3+y=z^2 \\
y^3+z=x^2 \\
z^3+x =y^2
\end{align*}
Find all pair $(x, y)$ in degrees such that
\begin{align*}
&\sin (x + 150^\circ) = \cos (y - 75^\circ), \\
&\cos x + \sin (y - 225^\circ) + \frac{\sqrt3}{2} = 0.
\end{align*}
For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations
$$x + py = n , \\ x + y = p^2$$
to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.
Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]
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 $T = x + y + z$
Solve the system of equations:
$$\begin{cases} \dfrac{xy}{x+y}=\dfrac{12}{5}\\ \\ \dfrac{yz}{y+z}=\dfrac{18}{5} \\ \\ \dfrac{zx}{z+y}=\dfrac{36}{13} \end{cases}$$
Find all solutions of the system consisting of $3$ equations:
$x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\
x_3 - x_4 = b \\
x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.
Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations:
\[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]
Find the real numbers $ x, y, z $ satisfying the system of equations
$$(z - x)(x - y) = a $$
$$(x - y)(y - z) = b$$
$$(y - z)(z - x) = c$$
where $ a, b, c $ are given real numbers.
Solve the following system of equations over the real numbers:
$2x_1 = x_5 ^2 - 23$
$4x_2 = x_1 ^2 + 7$
$6x_3 = x_2 ^2 + 14$
$8x_4 = x_3 ^2 + 23$
$10x_5 = x_4 ^2 + 34$
The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$?
$ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$
Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.
Find all real numbers $x,y,z$ satisfying the following system of equations:
\begin{align*}
xy+z&=-30\\
yz+x &= 30\\
zx+y &=-18
\end{align*}
Find natural $a, b, c, d$ that satisfy the system
$$\begin{cases} ab+cd=34
\\ ac-bd=19
\end{cases}$$
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds:
$ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\
x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\
x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\
x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\
x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\
x_6x_1(1\minus{}x_2)\equal{}x_3x_4$
Find all pairs $(x, y)$ of real numbers that satisfy the system of equations
$$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\
y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$