Found problems: 744
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
Show that the only integral solution to
\[\left\{ \begin{array}{l}
xy + yz + zx = 3n^2 - 1\\
x + y + z = 3n \\
\end{array} \right.
\]
with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
Solve the following system of equations:
$$\begin{cases}
x^3+y+z=1\\
x+y^3+z=1\\
x+y+z^3=1\end{cases}$$
Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$
Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.
Find all possible real values for $a, b$ and $c$ such that
(a) $a + b + c = 51$,
(b) $abc = 4000$,
(c) $0 < a \le 10$ and $c \ge 25$.
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.
Consider the following system of $10$ equations in $10$ real variables $v_1, \ldots, v_{10}$:
\[v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).\]
Find all $10$-tuples $(v_1, v_2, \ldots , v_{10})$ that are solutions of this system.
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations:
$$\begin{cases}
ab-c = 3 \\
a+bc = 4 \\
a^2+c^2 = 5\end{cases}$$
Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$
Solve the system of equations:
$ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$
$ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system
$$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\
a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\
a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$
has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\
16y^3 + 4y = 16x + 5 \end{cases}$
Let us denote by $<a>$ the distance from $a$ to the nearest integer. (For example, $<1,2> = 0.2$, $<\sqrt3> = 2-\sqrt3$) How many solutions does the system of equations have
$$\begin{cases} <19x>=y
\\ <96y>=x
\end{cases} \,\,\, ?$$
Given $x,y,z\in \mathbb{R^+}$. Find all sets of $x,y,z$ that correspond to $$x+y+z=x^2+y^2+z^2+18xyz=1$$
[i](Zhuge Liang)[/i]
Solve the system of equation
$$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$
Real numbers $a, b, c, x, y, z$ satisfy
$$\begin{aligned}
\begin{cases}
a^2+2bc=x^2+2yz,\\
b^2+2ca=y^2+2zx,\\
c^2+2ab=z^2+2xy.\\
\end{cases}
\end{aligned}$$
Prove that $a^2+b^2+c^2=x^2+y^2+z^2$.
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
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 $xy + 2yz + 3zx$.
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations:
$$\left \{\begin{matrix}
a\cdot b+c+d & = & 6, \\
b\cdot c+d+a & = & 2, \\
c\cdot d+a+b & = & 5, \\
d\cdot a+b+c & = & 3.
\end{matrix}\right .$$
Let $a_1, a_2, a_3$ be three distinct real numbers. Define
$$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\
b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\
b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$
Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$
When does equality hold?
Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$
Given the system of equations
\[ ax \plus{} (a \minus{} 1)y \equal{} 1 \\
(a \plus{} 1)x \minus{} ay \equal{} 1.
\]
For which one of the following values of $ a$ is there no solution $ x$ and $ y$?
$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ \minus{} 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt {2}$