MathDB

2

Part of 2017 VJIMC

Problems(2)

Integer sequence with iterated operation

Source: VJIMC 2017, Category I, Problem 2

4/2/2017
We say that we extend a finite sequence of positive integers (a1,…,an)(a_1,\dotsc,a_n) if we replace it by (1,2,…,a1−1,a1,1,2,…,a2−1,a2,1,2,…,a3−1,a3,…,1,2,…,an−1,an)(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n) i.e., each element kk of the original sequence is replaced by 1,2,…,k1,2,\dotsc,k. Géza takes the sequence (1,2,…,9)(1,2,\dotsc,9) and he extends it 20172017 times. Then he chooses randomly one element of the resulting sequence. What is the probability that the chosen element is 11?
Sequencesprobability
Increasing function implies existence of non-increasing one

Source: VJIMC 2017, Category II, Problem 2

4/2/2017
Prove or disprove the following statement. If g:(0,1)→(0,1)g:(0,1) \to (0,1) is an increasing function and satisfies g(x)>xg(x) > x for all x∈(0,1)x \in (0,1), then there exists a continuous function f:(0,1)→Rf:(0,1) \to \mathbb{R} satisfying f(x)<f(g(x))f(x) < f(g(x)) for all x∈(0,1)x \in (0,1), but ff is not an increasing function.
functionreal analysiscollege contests