1966 IMO Problems/Problem 1

Problem

In a mathematical contest, three problems, $A$ , $B$ , and $C$ were posed. Among the participants there were $25$ students who solved at least one problem each. Of all the contestants who did not solve problem $A$ , the number who solved $B$ was twice the number who solved $C$ . The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$ . How many students solved only problem $B$ ?

Solution

Let us draw a Venn Diagram.

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Let $a$ be the number of students solving both B and C. Then for some positive integer $x$ , $2x - a$ students solved B only, and $x - a$ students solved C only. Let $2y - 1$ be the number of students solving A; then $y$ is the number of students solving A only. We have by given $2y - 1 + 3x - a = 25$ and $y = 3x - 2a.$ Substituting for y into the first equation gives $9x - 5a = 26.$ Thus, because $x$ and $a$ are positive integers with $x-a \ge 0$ , we have $x = 4$ and $a = 2$ . (Note that $x = 9$ and $a = 11$ does not work.) Hence, the number of students solving B only is $2x - a = 8 - 2 = \boxed{6}.$

1966 IMO (Problems) • Resources
Preceded by First Question	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 2
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1966 IMO Problems/Problem 1

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