MathDB

Problem 3

Part of 2006 VJIMC

Problems(2)

replacing k by k+1 or 2k, don't get over n

Source: VJIMC 2006 1.3

6/28/2021
Two players play the following game: Let nn be a fixed integer greater than 11. Starting from number k=2k=2, each player has two possible moves: either replace the number kk by k+1k+1 or by 2k2k. The player who is forced to write a number greater than nn loses the game. Which player has a winning strategy for which nn?
gamecombinatorics
function and secant intersect

Source: VJIMC 2006 2.3

6/29/2021
For a function f:[0,1]→Rf:[0,1]\to\mathbb R the secant of ff at points a,b∈[0,1]a,b\in[0,1], a<ba<b, is the line in R2\mathbb R^2 passing through (a,f(a))(a,f(a)) and (b,f(b))(b,f(b)). A function is said to intersect its secant at a,ba,b if there exists a point c∈(a,b)c\in(a,b) such that (c,f(c))(c,f(c)) lies on the secant of ff at a,ba,b.
1. Find the set F\mathcal F of all continuous functions ff such that for any a,b∈[0,1]a,b\in[0,1], a<ba<b, the function ff intersects its secant at a,ba,b. 2. Does there exist a continuous function f∉Ff\notin\mathcal F such that for any rational a,b∈[0,1],a<ba,b\in[0,1],a<b, the function ff intersects its secant at a,ba,b?
function