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Problem 324

Combinatorics Difficulty 2.5 Multiple choice CEMC Fermat · Canada · 2026

Ava painted one picture each day for $31$ consecutive days and numbered the
paintings consecutively from $1$ to
$31$ . Three of the pictures that she
painted on Sundays have even numbers. On what day did she paint the
picture numbered 25?

Pick one

Official solution

If a Sunday occurs on day $n$ ,
then the next Sunday occurs on day $n+7$ ( $7$ days later).

Positive integers that differ by $7$
have different parity, meaning that one is even and the other
is odd.

Three of the pictures painted on Sundays have even numbers, and so the
last of these pictures must have been painted $2\times7=14$ days after the first was
painted. (Integers that differ by $14$ , or that differ by any even number,
have the same parity.)

If the picture numbered $2$ was the
first picture painted on a Sunday, then the pictures numbered $2+14=16$ and $16+14=30$ were also painted on
Sundays.

(You should confirm for yourself that this is the only possibility for
which three of the pictures painted on Sundays have even numbers.)

The picture numbered $30$ was painted
on a Sunday, and so the picture numbered $25$ , painted $5$ days earlier, was painted on a
Tuesday.

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