MathDB

3

Part of 2014 IberoAmerican

Problems(2)

Maximum sum on segments

Source:

9/24/2014
20142014 points are placed on a circumference. On each of the segments with end points on two of the 20142014 points is written a non-negative real number. For any convex polygon with vertices on some of the 20142014 points, the sum of the numbers written on their sides is less or equal than 11. Find the maximum possible value for the sum of all the written numbers.
combinatorics unsolvedcombinatorics
Iberoamerican Olympiad 2014, Problem 6

Source:

9/25/2014
Given a set XX and a function f:X→Xf: X \rightarrow X, for each x∈Xx \in X we define f1(x)=f(x)f^1(x)=f(x) and, for each j≥1j \ge 1, fj+1(x)=f(fj(x))f^{j+1}(x)=f(f^j(x)). We say that a∈Xa \in X is a fixed point of ff if f(a)=af(a)=a. For each x∈Rx \in \mathbb{R}, let π(x)\pi (x) be the quantity of positive primes lesser or equal to xx.
Given an positive integer nn, we say that f:{1,2,…,n}→{1,2,…,n}f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\} is catracha if ff(k)(k)=kf^{f(k)}(k)=k, for every k=1,2,…nk=1, 2, \dots n. Prove that:
(a) If ff is catracha, ff has at least π(n)−π(n)+1\pi (n) -\pi (\sqrt{n}) +1 fixed points. (b) If n≥36n \ge 36, there exists a catracha function ff with exactly π(n)−π(n)+1 \pi (n) -\pi (\sqrt{n}) + 1 fixed points.
functionalgorithminductioninequalitiesnumber theory unsolvednumber theory