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Problem 533

Algebra Difficulty 2.5 Multiple choice CEMC Cayley · Canada · 2023

Points $A$ , $B$ , $C$ , and $D$ are on a line in that order.

The distance from $A$ to $D$ is $24$ .

The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$ .

Point $C$ is halfway between $B$ and $D$ .

What is the distance from $A$ to
$C$ ?

Pick one

Official solution

Since $B$ is between $A$ and $D$ and $BD =
3AB $, then$ B $splits$ AD $in the ratio$ 1:3$.

Since $AD=24$ , then $AB = 6$ and $BD
= 18$.

Since $C$ is halfway between $B$ and $D$ , then $BC =
$\frac{1}{2}BD$ = 9$.

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Thus, $AC = AB + BC = 6 + 9 =
15$.

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