Olympiad problems

2024 Baltic Way, 20

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 $(a,b,c)$.

2022 Cyprus TST, 1

Tags: discriminant , algebra , number theory

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

1968 IMO, 3

Tags: polynomial , discriminant , algebra , system of equations

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.

1991 IMO Shortlist, 14

Tags: modular arithmetic , number theory , quadratic , perfect square , discriminant

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.$

1997 Akdeniz University MO, 1

Tags: number theory , discriminant

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.

2016 Philippine MO, 1

Tags: algebra , invariant , discriminant , quadratic

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? $(\text{a})$ $5x^2+5x-1$ $(\text{b})$ $x^2+6x+2$

1968 IMO Shortlist, 4

Tags: system of equations , discriminant , polynomial , algebra

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.

2022 Korea Junior Math Olympiad, 8

Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$