Found problems: 744
Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$
For which $k$ does the system $x^2 - y^2 = 0, (x-k)^2 + y^2 = 1$ have exactly:
(i) two,
(ii) three real solutions?
Determine all sets of real numbers $(x,y,z)$ which fulfills
$$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$
Solve the following system in the set of real numbers:
$x^2 -xy+y^2 = 7$,
$x^2y+xy^2 = -2$.
How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system?
$$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.
It's given system of equations
$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$
..........
$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$
such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations:
$$x^2 + y^2 + z^2 + w^2 = 4$$
$$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$
Solve system of equations
$$\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1
\\ x+\dfrac{x-y}{x^2+y^2}=2
\end{cases}$$
Find all complex solutions to the system
\begin{align*}
(a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\
(a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\
(1+i)a+2ic&=0.\end{align*}
Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1, x_2,...,, x_n$. Find all solutions for each such $n$.
$$\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}$$
Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\]
Solve the system of equations:
$$|\log_2(x + y)| + | \log_2(x - y)| = 3$$
$$xy = 3$$
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
$$\log_2\left({x \over yz}\right) = {1 \over 2}$$
$$\log_2\left({y \over xz}\right) = {1 \over 3}$$
$$\log_2\left({z \over xy}\right) = {1 \over 4}$$
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$
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 $q$.
Determine the real numbers $a, b, c, d$ so that
$$ab + c + d = 3, \,\, bc + d + a = 5, \,\, cd + a + b = 2 \,\,\,\, and \,\,\,\,da + b + c = 6$$
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that
\[(x+yt+z^2)(u+vt+wt^2)=1\]
For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$.
Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations:
\[ \{a\}+[b]+\{c\}=2,9\]\[ \{b\}+[c]+\{a\}=5,3\]\[\{c\}+[a]+\{b\}=4,0\]
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
Solve the system: $$\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}$$
Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
Find all real solutions $(x,y,z)$ of the system $$\begin{cases}\dfrac{4x^2}{1+4x^2}= y\\ \\\dfrac{4y^2}{1+4y^2}= z\\
\\ \dfrac{4z^2}{1+4z^2}= x \end{cases}$$
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
Find all real numbers $x, y, z$ that satisfy the following system
$$\sqrt{x^3 - y} = z - 1$$
$$\sqrt{y^3 - z} = x - 1$$
$$\sqrt{z^3 - x} = y - 1$$
Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\
b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\
c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\
d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$