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 6 of 6: Select the unique non-negative integer solution

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In plain words

Non-negativity of the remaining group size $d$ is what turns an equation with infinitely many real solutions into exactly one admissible integer solution.

b=6,\quad c=2

Detailed analysis

For $4b+c=26$ with $b,c\ge 0$ integers and $d=b-2c\ge0$ : $b=5$ forces $c=6$ but then $d=5-12<0$ , impossible; $b\ge7$ forces $c<0$ , also impossible; only $b=6,c=2$ gives a valid $d=2\ge0$ . Hence exactly $6$ students solved only $B$ .