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1

Part of 2013 AIME Problems

Problems(2)

The AIME Triathlon

Source: 2013 AIME I Problem 1

3/15/2013
The AIME Triathlon consists of a half-mile swim, a 3030-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
AMCAIME2013AIME I#1algebra
Metric timekeeping

Source: AIME II 2013, Problem 1

4/4/2013
Suppose that the measurement of time during the day is converted to the metric system so that each day has 1010 metric hours, and each metric hour has 100100 metric minutes. Digital clocks would then be produced that would read 9:999{:}99 just before midnight, 0:000{:}00 at midnight, 1:251{:}25 at the former 3:003{:}00 \textsc{am}, and 7:507{:}50 at the former 6:006{:}00 \textsc{pm}. After the conversion, a person who wanted to wake up at the equivalent of the former 6:366{:}36 \textsc{am} would have to set his new digital alarm clock for A:BC\text{A:BC}, where A\text{A}, B\text{B}, and C\text{C} are digits. Find 100A+10B+C100\text{A} + 10\text{B} + \text{C}.
ratiofunctionAMCAIMEnumber theory