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2023 SMT Guts Round 1 p1-3 - Stanford Math Tournament

Source:

August 31, 2023
combinatoricsalgebraStanford Math Tournament

Problem Statement

p1. To convert between Fahrenheit, FF, and Celsius, CC, the formula is F=95C+32F = \frac95 C + 32. Jennifer, having no time to be this precise, instead approximates the temperature of Fahrenheit, F^\widehat F, as F^=2C+30\widehat F = 2C + 30. There is a range of temperatures C1≤C≤C2C_1 \le C \le C_2 such that for any CC in this range, ∣F^−F∣≤5| \widehat F - F| \le 5. Compute the ordered pair (C1,C2)(C_1,C_2).
p2. Compute integer xx such that x23=27368747340080916343x^{23} = 27368747340080916343.
p3. The number of ways to flip nn fair coins such that there are no three heads in a row can be expressed with the recurrence relation S(n+1)=a0S(n)+a1S(n−1)+...+akS(n−k) S(n + 1) = a_0 S(n) + a_1 S(n - 1) + ... + a_k S(n - k) for sufficiently large nn and kk where S(n)S(n) is the number of valid sequences of length nn. What is ∑n=0k∣an∣\sum^k_{n=0}|a_n|?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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