MathDB

3

Part of ICMC 7

Problems(2)

Encryprion keys on Mathsenger

Source: ICMC 7 Round 1 Problem 3

1/8/2024
There are 105 users on the social media platform Mathsenger, every pair of which has a direct messaging channel. Prove that each messaging channel may be assigned one of 100 encryption keys, such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key.
Proposed by Fredy Yip
combinatoricsICMC
Expected value with polynomials

Source: ICMC 7 Round 2 Problem 3

3/12/2024
Let NN{} be a fixed positive integer, SS{} be the set {1,2,…,N}\{1, 2,\ldots , N\} and F\mathcal{F} be the set of functions f:S→Sf:S\to S such that f(i)⩾if(i)\geqslant i for all i∈S.i\in S. For each f∈Ff\in\mathcal{F} let PfP_f be the unique polynomial of degree less than NN{} satisfying Pf(i)=f(i)P_f(i) = f(i) for all i∈S.i\in S. If ff{} is chosen uniformly at random from F\mathcal{F} determine the expected value of Pf′(0)P_f'(0) wherePf′(0)=dPf(x)dx∣x=0.P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.Proposed by Ishan Nath
polynomialreal analysisexpected value