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 5 of 6: Combine into one equation in

b,c

Previous step Next step

In plain words

Three equations in five unknowns collapse to one equation in $b,c$ once $a$ , $d$ , and $e$ are eliminated by substitution.

4b+c=26

Detailed analysis

Substituting $e=a-1$ into the total gives $2a+b+c+d=26$ . Using $a=b+c$ turns this into $3b+3c+d=26$ , and substituting $d=b-2c$ from step 2 gives $4b+c=26$ .