2025 USAJMO Problems/Problem 4

Problem

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that $\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.$ Note: $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$ .

Solution

By Vandermonde's, $\frac 12\binom{\sum a_i}2=\frac 12\left(\sum\binom{a_i}2+\sum_\text{cyc}a_ia_j\right)\ge\frac 12\sum ia_i^2\ge\sum i\binom{a_i}2,$ with equality at $a_0=0,1$ and $a_i=0$ for $i>0.~\square$

~rhydon516 (sol credits to leo)

2025 USAJMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
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2025 USAJMO Problems/Problem 4

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