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Problem 106

Combinatorics Difficulty 1.3 Multiple choice CEMC Fermat · Canada · 2014

In a group of five friends:

Amy is taller than Carla.
Dan is shorter than Eric but taller than Bob.
Eric is shorter than Carla.

Who is the shortest?

Pick one

Official solutions — 2

We use $A,B,C,D,E$ to represent Amy, Bob, Carla, Dan, and Eric, respectively.

We use the greater than symbol ( $>$ ) to represent “is taller than" and the less than symbol ( $<$ ) to represent “is shorter than".

From the first bullet, $A>C$ .

From the second bullet, $D<E$ and $D>B$ so $E>D>B$ .

From the third bullet, $E<C$ or $C>E$ .

Since $A>C$ and $C>E$ and $E>D>B$ , then $A>C>E>D>B$ , which means that Bob is the shortest.

We use $A,B,C,D,E$ to represent Amy, Bob, Carl, Dan, and Eric, respectively.

We use the greater than symbol ( $>$ ) to represent “is taller than" and the less than symbol ( $<$ ) to represent “is shorter than".

From the first bullet, $A>C$ .

From the second bullet, $D<E$ and $D>B$ so $E>D>B$ .

From the third bullet, $E<C$ or $C>E$ .

Since $A>C$ and $C>E$ and $E>D>B$ , then $A>C>E>D>B$ , which means that Bob is the shortest.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.