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, 2003

Algebra Difficulty 6.7 National Olympiad Prove it Italy

Problem:

Little Marco goes up and down from one floor to another on the escalator of a shopping center, one step at a time. If he proceeds in the direction of motion of the escalator at a constant speed relative to it (that is, the time interval between one step and the next is constant), he steps on 15 steps; if he proceeds in the opposite direction (with the same time interval between one step and the next) he steps on 35 of them. How many steps would Marco step on in going from one floor to the other if the escalator were stopped?

Solution

Solution:

The answer is 21. Let us denote by $V_{s}$ the speed of the escalator, in steps per unit of time, and by $V_{m}$ the speed of Marco, in the same unit of measure. If $T_{1}$ and $T_{2}$ are, respectively, the times taken by Marco to travel the escalator in the concordant direction and in the discordant direction, then the following relations hold
$\begin{array}{r} \left(V_{m}+V_{s}\right) T_{1}=N \\ \left(V_{m}-V_{s}\right) T_{2}=N \\ V_{m} \cdot T_{1}=15 \\ V_{m} \cdot T_{2}=35 \end{array}$
where $N$ is the number of steps that Marco would step on if the escalator were stopped. From formulas (1) and (2) it can be deduced that
$\frac{T_{1}}{T_{2}}=\frac{N-15}{35-N}$
while from formulas (3) and (4), it can be deduced that
$\frac{T_{1}}{T_{2}}=\frac{15}{35}$
Putting together the two relations (5) and (6) thus obtained, one gets
$N=\frac{2 \cdot 15 \cdot 35}{15+35}=21$
which is exactly the harmonic mean of the numbers 15 and 35.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.