Found problems: 744
Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations:
$$x^2 - y = (z -1)^2 $$
$$y^2 - z = (x -1)^2$$
$$z^2 - x = (y - 1)^2$$
Solve the system of equations
$$\begin{cases} x^2+ [y]=10
\\ y^2+[x]=13
\end{cases}$$
($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).
Solve in $ \mathbb{R}^3 $ the system:
$$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$
[i]Cristinel Mortici[/i]
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system:
$$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$
Determine all triples $(a,b,c)$ of real numbers with
\[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]
Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.
\[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw
$$x + y + z = a,$$
$$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$
then at least one of them is equal to $ a $.
Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$
Find all possible values of $a$.
Real numbers $x,y,a$ are such that $x+y=x^2+y^2=x^3+y^3=a$. Find all the possible values of $a$.
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
Find all rational solutions of
\[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]
Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:
\[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
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?
Solve the system of equations:
$$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$
Solve the following system of equations:
$$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$
Solve the following system for real $x,y,z$
\[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]
Solve the following system of equations in integers:
$\begin{cases}
x^2+2xy+8z=4z^2+4y+8\\
x^2+y+2z=156 \\
\end{cases}$
Find the value of $a$ satisfying
\begin{align*}
a+b&=3\\
b+c&=11\\
c+a&=61
\end{align*}
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$
$$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$