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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/3/e/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 3 / e / Solution by

Codex  0 2026-10-25

Prove the stronger statement with a nonempty subset. Order the

n \geq 1

a_{1}, \dots, a_{n}

, and form the prefix sums

s_{0} = 0, s_{j} = \sum_{k = 1}^{j} a_{k} (1 \leq j \leq n) .

There are

n + 1

such sums but only

n

residue classes modulo

n

. The pigeonhole principle gives indices

0 \leq i < j \leq n

with

s_{i} \equiv s_{j} (mod n)

. Their difference satisfies

s_{j} - s_{i} = a_{i + 1} + \dots + a_{j} \equiv 0 (mod n) .

The selected elements form a nonempty subset of

S

, proving the claim. Neither positivity nor distinct values in a general ordered sequence are needed for the consecutive sum divisible by the sequence length argument; distinctness merely makes the present input a set.

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