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Problem 396

Combinatorics Difficulty 2.6 Find the answer CEMC Gauss (Grade 8) · Canada · 2019

Kathy owns more cats than Alice and more dogs than Bruce. Alice owns more dogs than Kathy and fewer cats than Bruce. Which of the statements must be true?

$\text{Bruce owns the fewest cats.}$
$\text{Bruce owns the most cats.}$
$\text{Kathy owns the most cats.}$
$\text{Alice owns the most dogs.}$
$\text{Kathy owns the fewest dogs.}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solutions — 2

Solution 1

We begin by separating the given information, as follows:

Kathy owns more cats than Alice
Kathy owns more dogs than Bruce
Alice owns more dogs than Kathy
Bruce owns more cats than Alice

From bullets 2 and 3, we can conclude that Alice owns more dogs than both Kathy and Bruce.

From bullet 4, we can conclude that answer (A) is not true.

From bullets 1 and 4, we can conclude that both Kathy and Bruce own more cats than Alice.

However, we cannot determine if Kathy owns more cats than Bruce, or vice versa.

Therefore, we cannot conclude that (B) or (C) must be true.

From bullet 2, we can conclude that (E) is not true.

Thus the statement which must be true is (D).

Solution 2

The area of the rectangle with length $x$ and width $y$ is $x\times y$ .

The area of the triangle with base 16 and height $x$ is $\frac12\times 16\times x$ or $8\times x$ .

The area of the rectangle is equal to the area of the triangle, or $x\times y=8\times x$ , and so $y=8$ .

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