Found problems: 8
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$
Positive integers $a$, $b$ and $c$ satisfy the system of equations
\begin{align*}
(ab-1)^2&=c(a^2+b^2)+ab+1,\\
a^2+b^2&=c^2+ab.
\end{align*}
a) Prove that $c+1$ is a perfect square.
b) Find all such triples $(a,b,c)$.
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
Let $m \in {\mathbb R}$ and
$$x^2+(m-4)x+(m^2-3m+3)=0$$
equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.
Find all pairs of integers $(m, n)$ which satisfy the equation
\[(2n^2+5m-5n-mn)^2=m^3n\]
Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
The operations below can be applied on any expression of the form \(ax^2+bx+c\).
$(\text{I})$ If \(c \neq 0\), replace \(a\) by \(4a-\frac{3}{c}\) and \(c\) by \(\frac{c}{4}\).
$(\text{II})$ If \(a \neq 0\), replace \(a\) by \(-\frac{a}{2}\) and \(c\) by \(-2c+\frac{3}{a}\).
$(\text{III}_t)$ Replace \(x\) by \(x-t\), where \(t\) is an integer. (Different values of \(t\) can be used.)
Is it possible to transform \(x^2-x-6\) into each of the following by applying some sequence of the above operations?
$(\text{a})$ \(5x^2+5x-1\)
$(\text{b})$ \(x^2+6x+2\)