Found problems: 744
Solve the system $\begin{cases} x^3 - y^3 = 26 \\
x^2y - xy^2 = 6
\end{cases}$ in $C$
[hide=other version]solved below
Solve the system $\begin{cases} x^3 - y^3 = 2b \\
x^2y - xy^2 = b
\end{cases}$[/hide]
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
\[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]
Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying
$$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$
Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.
Determine all real numbers $a, b, c, d$ for which
$$ab + c + d = 3$$
$$bc + d + a = 5$$
$$cd + a + b = 2$$
$$da + b + c = 6$$
Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$
Find all three real numbers $(x, y, z)$ satisfying the system of equations $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}$$ $$x^2 + y^2 + z^2 = xy + yz + zx + 4$$
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
Solve the equation system for real numbers:
$$x_1+x_2=x_3^2$$
$$x_2+x_3=x_4^2$$
$$x_3+x_4=x_1^2$$
$$x_4+x_1=x_2^2$$
Let $n \ge 2$ be an integer. Consider the system of equations
\begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align}
1. Prove that $(1)$ has infinitely many real solutions $(x_1,\dotsc,x_n)$ such that the numbers $x_1,\dotsc,x_n$ are distinct.
2. Prove that every solution of $(1)$, such that the numbers $x_1,\dotsc,x_n$ are not all equal, satisfies $\vert x_1x_2\cdots x_n\vert=2^{n/2}$.
Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations
$\begin{cases} x^2 - y = (z - 1)^2\\
y^2 - z = (x - 1)^2 \\
z^2 - x = (y -1)^2 \end{cases}$.
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
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 $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\
... \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$
Find $3$ numbers, each of which is equal to the square of the difference of the other two.
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 $a^2+b^2+c^2+d^2$.
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations
\[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.
Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.
Six integers $a,b,c,d,e,f$ satisfy:
$\begin{cases}
ace+3ebd-3bcf+3adf=5 \\
bce+acf-ade+3bdf=2
\end{cases}$
Find all possible values of $abcde$
[i]D. Bazyleu[/i]
Solve the system of equations
$$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$
Solve the system of equations
$$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$
Find the values of the real numbers $x,y,z$ that correspond to the system of equations.
$$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$
$$xy + yz + zx=1$$
[i](Heir of Ramanujan)[/i]
Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\
x + y = b \end{cases}$ in $C$