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Problem 380

Combinatorics Difficulty 2.6 Find the answer CEMC Gauss (Grade 8) · Canada · 2017

On coach Wooden’s basketball team:

Meghan is the tallest player,
Meghan’s height is 188 cm, and
Avery is the shortest player.

When used with the information above, which of the following single statements is enough to determine Avery’s height?

$\text{The median of the players' heights is 170 cm}$
$\text{The mode of the players' heights is 160 cm}$
$\text{The mean of the players' heights is 165 cm}$
$\text{The range of the players' heights is 33 cm}$
$\text{There are 10 players on the team}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solution 1

Since 78 is 2 less than 80 and 82 is 2 greater than 80, the mean of 78 and 82 is 80.

Since the mean of all four integers is 80, then the mean of 83 and $x$ must also equal 80.

The integer 83 is 3 greater than 80, and so $x$ must be 3 less than 80.

That is, $x=80-3=77$ .

(We may check that the mean of $78,83,82,$ and $77$ is indeed 80.)

Solution 2

Since the mean of the four integers is 80, then the sum of the four integers is $4\times80=320$ .

Since the sum of 78, 83 and 82 is 243, then $x=320-243=77$ .

Therefore, $x$ is 77 which is 3 less than the mean 80.

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