Problem 1

Three problems

A

B

, and

C

were given on a mathematics olympiad. All 25 students solved at least one of these problems. The number of students who solved

B

and not

A

is twice the number of students who solved

C

and not

A

. The number of students who solved only

A

is greater by 1 than the number of students who, along with

A

, solved at least one other problem. Among the students who solved only one problem, half solved

A

. How many students solved only

B

Step 1 of 6: Name the five disjoint solving groups

Previous step Next step

In plain words

A Venn diagram of $A,B,C$ has seven regions; merging the regions that overlap $A$ into a single variable $e$ keeps the bookkeeping small while still capturing every condition given.

a+b+c+d+e=25

Detailed analysis

Let $a,b,c$ be the numbers of students who solved only $A$ , only $B$ , only $C$ respectively, let $d$ be the number who solved $B$ and $C$ but not $A$ , and let $e$ be the number who solved $A$ together with at least one other problem. These five groups partition all 25 students, so $a+b+c+d+e=25$ .