Found problems: 744
Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right),
\\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right).
\end{align*}
Solve the system of equations
$$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$
Solve the system of equations:
$$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\
\log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$
Solve the system of equations
$$
\begin{array}{l}<br />
x^2y^2 + x^2z^2 = axyz\\<br />
y^2z^2 + y^2x^2 = bxyz\\<br />
z^2x^2 + z^2y^2 = cxyz.<br />
\end{array}$$
Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\
x^4 + y^4 + x^2y^2 = 8 \end{cases}$
The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is:
(A): $46$, (B): $48$, (C): $50$, (D): $52$, (E) None of the above.
If $a$ and $b$ are real numbers such that $$\begin{cases} a^3-3ab^2 = 8 \\ b^3-3a^2b = 11 \end{cases}$$ then what is $a^2+b^2$?
Find $x+y+z$ when $$a_1x+a_2y+a_3z= a$$$$b_1x+b_2y+b_3z=b$$$$c_1x+c_2y+c_3z=c$$ Given that $$a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9$$$$a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17$$$$a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c-bc_1\right)=-8$$$$a_1\left(b_2c-bc_2\right)-a_2\left(b_1c-bc_1\right)+a\left(b_1c_2-b_2c_1\right)=7.$$
[i]2017 CCA Math Bonanza Lightning Round #5.1[/i]
Solve the system of equations in real numbers:
\[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]
Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.
Solve the system of equations
$x+y+z=\pi$
$\tan x\tan z=2$
$\tan y\tan z=18$
Solve in the nonnegative real numbers the system of equations
$$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\
x_0 = x_{1999} \end{cases}$$
Determine all solutions of the system
$$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$
in which the first three are equations and the last one is a linear inequality.
Find all pairs of real numbers $a,b$ for which the system of equations $$ \begin{cases} \dfrac{x+y}{x^2 +y^2} = a \\ \\ \dfrac{x^3 +y^3}{x^2 +y^2} = b \end{cases}$$ has a real solution.
For each integer $n \ge 2$, find all integer solutions of the following system of equations:
\[x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}\]
\[x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}\]
\[\vdots\]
\[x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}\]
If the reals $a,b,c,d,e,f,g,h,i$ satisfy
$$\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}$$show that $cf+fi+ic=3$ holds as well.
Solve the system of equations: $\begin{cases}
x_1 + x_2 + x_3 = 6 \\
x_2 + x_3 + x_4 = 9 \\
x_3 + x_4 + x_5 = 3 \\
x_4 + x_5 + x_6 = -3 \\
x_5 + x_6 + x_7 = -9 \\
x_6 + x_7 + x_8 = -6 \\
x_7 + x_8 + x_1 = -2 \\
x_8 + x_1 + x_2 = 2 \end{cases}$
Find all pairs of real numbers $x$ and $y$ which satisfy the following equations:
\begin{align*}
x^2 + y^2 - 48x - 29y + 714 & = 0 \\
2xy - 29x - 48y + 756 & = 0
\end{align*}
For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?
If $x,y,z$ are real numbers satisfying relations
\[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\]
prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\
x_2x_3(3a-2x_4) = a^3\\
...\\
x_{n-2}x_{n-1}(3a-2x_n) = a^3\\
x_{n-1}x_n(3a-2x_1) = a^3 \\
x_nx_1(3a-2x_2) = a^3 \end {cases}$
.
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[
x^{2}-yz-zu-yu=a\]
\[
y^{2}-zu-ux-xz=b\]
\[
z^{2}-ux-xy-yu=c\]
\[
u^{2}-xy-yz-zx=d\]
Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations
$$\begin{cases}
x+y+z=n\\
xyz = 2t^3.
\end{cases}$$
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.