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Problem 524

Combinatorics Difficulty 2.5 Multiple choice CEMC Gauss (Grade 8) · Canada · 2023

The numbers 41, 35, 19, 9, 26, 45, 13, 28 are arranged in pairs
so that the sum of the numbers in each pair is the same. The number
paired with 13 is

Pick one

Official solution

The sum of $P$ and $Q$ is equal to 5, and so $P$ and $Q$ are (in some order) either equal to 1
and 4, or they are equal to 2 and 3.

The difference between $R$ and $S$ is equal to 5, and so $R$ and $S$ must be (in some order) equal to 1 and
6.

Since $R$ and $S$ are equal to 1 and 6, then neither
$P$ nor $Q$ can be 1, which means that $P$ and $Q$ cannot be equal to 1 and 4, and so they
must be equal to 2 and 3.

The only numbers from 1 to 6 not accounted for are 4 and 5.

Since $T$ is greater than $U$ , then the number that replaces $T$ is 5.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.