Found problems: 744
The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]
Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that
\[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]
Solve the system of equations:
$$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.
Solve the system of equations
$$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
Solve the following system of equations for real $x,y$ and $z$:
\begin{eqnarray*}
x &=& \sqrt{2y+3}\\
y &=& \sqrt{2z+3}\\
z &=& \sqrt{2x+3}.
\end{eqnarray*}
Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
Solve in positives $$x^y=z,y^z=x,z^x=y$$
Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.
Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .
Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations:
$ x_1\plus{}ax_2\equal{}0$
$ x_2\plus{}a^2x_3\equal{}0$
…
$ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$
…
$ x_n\plus{}a^nx_1\equal{}0$
Solve the system
\begin{align*}
36x^2y-27y^3&~=~8,\\
4x^3-27xy^2&~=~4.\end{align*}
Find all pairs (x; y) of real numbers satisfying the system of equations
$x^3 + 1 - xy^2 - y^2 = 0$;
$y^3 - 1 - x^2y + x^2 = 0$.
Darij
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$
Find all real values of the parameter $a$ for which the system of equations
\[x^4 = yz - x^2 + a,\]
\[y^4 = zx - y^2 + a,\]
\[z^4 = xy - z^2 + a,\]
has at most one real solution.
Find all real solutions of the system of equations
$$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\
(x_4 + x_5 + x_1)^5 = 3x_2\\
(x_5 + x _1 + x_2)^5 = 3x_3\\
(x_1 + x_2 + x_3)^5 = 3x_4\\
(x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$
(L. Tumescu , Romania)
Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations
$$a^3 - 2a = abc - b - c$$
$$b^3 - 2b = bca - c - a$$
$$c^3 - 2c = cab - a - b.$$
For real numbers $x$, $y$ and $z$, solve the system of equations:
$$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$
If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is:
$ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations
$2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$
$x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$
$x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$
have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.
How many positive integer solutions does the following system of equations have?
$$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\
\end{cases}$$
Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.
Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$):
\[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]