Team Selection Test #0im5

Team Selection TestHard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that

a_1 + \dots + a_i \le b_1 + \dots + b_i \quad \text{for every } i = 1, \dots, n-1

and

a_1 + \dots + a_n = b_1 + \dots + b_n.

Suppose that for any real number $m$ , the number of pairs $(i, j)$ with $a_i - a_j = m$ equals the number of pairs $(k, \ell)$ with $b_k - b_\ell = m$ . Prove that $a_i = b_i$ for $i = 1, \dots, n$ .

Note: It is important to interpret the condition that for any real number $m$ , the number of pairs $(i, j)$ with $a_i - a_j = m$ equals the number of pairs $(k, \ell)$ with $b_k - b_\ell = m$ . It means that we have two identical multi-sets (a multi-set is a set that allows repeated elements)