Track / Stage 4 / 124 of 340 #384 of 2000

Problem 384

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

3-4. Two friends need to get to the neighboring city. They have one bicycle that only one person can ride at a time. What is the minimum time it will take for both to reach the city (considering the last one to arrive), if their walking speeds are $u_{1}$ and $u_{2}$ , their cycling speeds are $v_{1}$ and $v_{2}$ , and the distance to the city is $S$ (they can return and leave the bicycle for each other)?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solve 4.

Part 5. Does there exist a pentagon with sides of 3, 4, 9, 11, and 13 cm, into which a circle can be inscribed?

Solve 5.

\section*{8th grade}

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