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BMT 2022 Guts Round Set 7 p19-21

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March 18, 2024
algebracombinatoricsgeometry

Problem Statement

Guts Round / Set 7
p19. Let N≥3N \ge 3 be the answer to Problem 21.
A regular NN-gon is inscribed in a circle of radius 11. Let DD be the set of diagonals, where we include all sides as diagonals. Then, let D′D' be the set of all unordered pairs of distinct diagonals in DD. Compute the sum ∑{d,d′}∈D′ℓ(d)2ℓ(d′)2,\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,where ℓ(d)\ell (d) denotes the length of diagonal dd.
p20. Let NN be the answer to Problem 1919, and let MM be the last digit of NN.
Let ω\omega be a primitive MMth root of unity, and define P(x)P(x) such thatP(x)=∏k=1M(x−ωik),P(x) = \prod^M_{k=1} (x - \omega^{i_k}),where the iki_k are chosen independently and uniformly at random from the range {0,1,...,M−1}\{0, 1, . . . ,M-1\}. Compute E[P(⌋1250N⌋)].E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].
p21. Let NN be the answer to Problem 2020.
Define the polynomial f(x)=x34+x33+x32+...+x+1f(x) = x^{34} +x^{33} +x^{32} +...+x+1. Compute the number of primes p<Np < N such that there exists an integer kk with f(k)f(k) divisible by pp.
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