Competition problem — Geometry — Maths Olympiad Prep

Library / /9 of 520

Geometry Difficulty 4.5 AIME Find the answer

8. Given that the perimeter of the regular pentagon square $A B C D E$ is $2000 \mathrm{~m}$ , two people, A and B, start from points $A$ and $C$ respectively at the same time, walking around the square in the direction of $A \rightarrow B \rightarrow C \rightarrow D \rightarrow E \rightarrow A \rightarrow \cdots$ . Person A's speed is $50 \mathrm{~m} / \mathrm{min}$ , and person B's speed is $46 \mathrm{~m} / \mathrm{min}$ . Then, after $\qquad$ $\min$ , A and B will first start walking on the same side.

Your answer

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

Solution

8. 104 .

Suppose that when person A has walked $x$ edges, A and B start walking on the same edge for the first time. At this point, A has walked $400 x \mathrm{~m}$ , and B has walked $46 \times$ $\frac{400 x}{50}=368 x \mathrm{~m}$ . Therefore,
$368(x-1)+800-400(x-1)>400 \text {, }$

and $(368 x+800)-400 x \leqslant 400$ .
Thus, $12.5 \leqslant x<13.5$ .
Hence, $x=13$ . At this point, $t=\frac{400 \times 13}{50}=104$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.