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Problem 364

Combinatorics Difficulty 2.5 Multiple choice CEMC Gauss (Grade 7) · Canada · 2023

Each number from 1 to 6 replaces one of the letters $P, Q, R, S, T$ , and $U$ . The sum of $P$ and $Q$ is 5 and the difference between $R$ and $S$ is 5. If $T$ is greater than $U$ , what number replaces the letter $T$ ?

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.