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 4 of 6: Translate the "half solved A" condition

In plain words

"Solved only one problem" is the union of three disjoint regions, so halving it is a statement about their sum.

a=b+c

Detailed analysis

Students who solved only one problem number $a+b+c$ ; half of them solved $A$ means $a=\frac{1}{2}(a+b+c)$ , i.e. $a=b+c$ .