MathDB

5

Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(3)

Iteration of a weird function

Source: St. Petersburg 2023 11.5

8/12/2023
Let a>1a>1 be a positive integer and let f(n)=n+[a{n2}]f(n)=n+[a\{n\sqrt{2}\}]. Show that there exists a positive integer nn, such that f(f(n))=f(n)f(f(n))=f(n), but f(n)≠nf(n) \neq n.
functionalgebra
Expression attains positive and negative values

Source: St. Petersburg 2023 10.5

8/12/2023
Let x0,x1,…,xn−1,xn=x0x_0, x_1, \ldots, x_{n-1}, x_n=x_0 be reals and let f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} be a function. The numbers yiy_i for i=0,1,…,n−1i=0,1, \ldots, n-1 are chosen such that yiy_i is between xix_i and xi+1x_{i+1}. Prove that ∑i=0n−1(xi+1−xi)f(yi)\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i) can attain both positive and negative values, by varying the yiy_i.
algebra
kn starts with d

Source: St. Petersburg 2023 9.5

8/12/2023
For a positive integer nn and a nonzero digit dd, let f(n,d)f(n, d) be the smallest positive integer kk, such that knkn starts with dd. What is the maximal value of f(n,d)f(n, d), over all positive integers nn and nonzero digits dd?
algebra