Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 109 / 2 / a / Solution

For a sequence $J$ in ${\mathbb{F}}_p^2$, write $(r\mid J)$ for the number of $r$-term subsequences whose sum is zero. We use the following consequence of the Chevalley-Warning theorem: if $|J|=3p-1$, thenIndeed, with one variable $x_i$ for each term $(a_i,b_i)$, apply Chevalley--Warning toTheir degree sum is $3(p-1)<3p-1$. A common zero has support of size $0,p$, or $2p$ modulo $p$, and each fixed support contributes $(p-1{)}^{|S|}$ assignments. Reducing modulo $p$ gives the displayed congruence.

Now let the given $3p$ terms have total sum zero. If no $p$ of them summed to zero, delete any one term and apply the congruence to the remaining $3p-1$ terms. It givesso those remaining terms contain a zero-sum $2p$-subsequence. Its complement in the original $3p$ terms has size $p$ and sum zero, contradicting the assumption. Therefore the required $p$ terms exist.