Question

s. Linda ha sti wecks to prcpare for an cvan and budlgets 30 hours to study, She studie every day for at least an hour, and only studies a whole number of hours each day. Prove that for some period of consecutive days, she will regardless of her schedule.

Answer 1

6 weeks is 42 days let x_1, x_2 ... x_42 be the number of hours studied on that day consider the 42 sums x_1, x_1 + x_2, x_1 + x_2 + x_3, ... (x_1 + x_2 + x_3 ... + x_n + 2) since these are 42, the will exist 2 numbers that are equivalent modulo 33 via region - hole principle say (x_1 + x_2 ... + x_k) identicalto (x_1 + x_2 ... + x_j) for j < k then (x_k + 1 + x_k + 2 + ... x_k) identicalto 0 (33) as sigma x_i = 50 the only such number is 33 so there must exist some continuous days such that, sum of number of hours worked for these days is 33