Rates and Work (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Rates and Work: core idea

Olympiad modeling.

**(Proof track.)** Give a complete justification. A pipe fills in $8$ h; a drain empties in $12$ h. Starting empty with both open, find fill time.

Hint: Define a consistent unit and derive the equation from rates or conserved quantities. Answer: 24.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. A boat's downstream/upstream speeds are $18,10$ . Find still-water speed.
2. **(Proof track.)** Give a complete justification. Two trains $300$ mi apart travel toward each other at $55,65$ mph. Find meeting time.
3. **(Proof track.)** Give a complete justification. A $30\%$ solution is mixed with $50\%$ solution to get $18$ L of $40\%$ . Find liters of $30\%$ .
4. **(Proof track.)** Give a complete justification. A worker completes $3/8$ of a job in $5$ days. Find total days at that rate.
5. **(Proof track.)** Give a complete justification. A car travels $x$ miles at $40$ mph and $x$ miles at $60$ mph. Find average speed.
6. **(Proof track.)** Give a complete justification. Two faucets fill in $9,18$ h; after 2 h the slower is closed. Find total time.
7. **(Proof track.)** Give a complete justification. A population increases $20\%$ then decreases $20\%$ . Starting at $500$ , find final population.
8. **(Proof track.)** Give a complete justification. A runner at $6$ mph is chased after 15 minutes by a cyclist at $15$ mph. Find cyclist travel time to catch.
9. **(Proof track.)** Give a complete justification. An alloy is $60\%$ copper. How much pure copper added to $25$ kg makes it $75\%$ ?