MathDB

4

Part of 2023 European Mathematical Cup

Problems(2)

Sweet and Smooth!

Source: EMC 2023 Juniors P4

12/18/2023
We say that a 20232023-tuple of nonnegative integers (a_1,\hdots,a_{2023}) is sweet if the following conditions hold:
[*] a_1+\hdots+a_{2023}=2023 [*] \frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1
Determine the greatest positive integer LL so that a_1+2a_2+\hdots+2023a_{2023}\ge L holds for every sweet 20232023-tuple (a_1,\hdots,a_{2023})
Ivan Novak
emcalgebraInequality2023
Conditional squares imply injectivity

Source: EMC 2023 Seniors P4

12/18/2023
Let f ⁣:N→Nf\colon\mathbb{N}\rightarrow\mathbb{N} be a function such that for all positive integers xx and yy, the number f(x)+yf(x)+y is a perfect square if and only if x+f(y)x+f(y) is a perfect square. Prove that ff is injective.
Remark. A function f ⁣:N→Nf\colon\mathbb{N}\rightarrow\mathbb{N} is injective if for all pairs (x,y)(x,y) of distinct positive integers, f(x)≠f(y)f(x)\neq f(y) holds.
Ivan Novak
EMC 20232023functional equationfunctionIvan Novak orzevan orz