Olympiad problems

2021 Romanian Master of Mathematics Shortlist, A1

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \] for all real numbers $x$ and $y$.

2022 European Mathematical Cup, 3

Tags: substitution , cauchy functional equation , functional equation , function , algebra

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$ for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.

2024 Bangladesh Mathematical Olympiad, P6

Tags: algebra , substitution , polynomial

Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.

2020 Dürer Math Competition (First Round), P1

Tags: algebra , substitution

In maths class Albrecht had to compute $(a+2b-3)^2$ . His result was $a^2 +4b^2-9$ . ‘This is not correct’ said his teacher, ‘try substituting positive integers for $a$ and $b$.’ Albrecht did so, but his result proved to be correct. What numbers could he substitute? a) Show a good substitution. b) Give all the pairs that Albrecht could substitute and prove that there are no more.

2019 Ramnicean Hope, 1

Tags: riemann sum , calculus , real analysis , definite integral , substitution

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]