Mathematica competitions in Croatia

Let $n$ be the number of students.

Initially, the $960$ tokens are distributed equally, so each student gets $\frac{960}{n}$ tokens.

When the teacher arrives, each student gives him $4$ tokens, so the teacher receives $4n$ tokens in total. Now there are $n+1$ people (the $n$ students and the teacher), and the $960$ tokens are distributed among them equally, so each person gets $\frac{960}{n+1}$ tokens.

After giving away $4$ tokens, each student has: $$\frac{960}{n}-4$$ tokens.

The teacher has $4n$ tokens.

Since now everyone has the same number of tokens: $$\frac{960}{n}-4=\frac{960}{n+1}$$

Multiply both sides by $n(n+1)$: $$(n+1)\cdot 960-4n(n+1)=n\cdot 960$$

Expand: $$960n+960-4n^2-4n=960n$$

Subtract $960n$ from both sides: $$960-4n^2-4n=0$$

Bring all terms to one side: $$4n^2+4n-960=0$$

Divide both sides by $4$: $$n^2+n-240=0$$

Solve the quadratic equation: $$n=\frac{-1\pm \sqrt{1+4\cdot 240}}{2}=\frac{-1\pm \sqrt{961}}{2}=\frac{-1\pm 31}{2}$$

So $n=\frac{30}{2}=15$ or $n=\frac{-32}{2}=-16$.

Since $n$ must be positive, $n=15$.

Answer: $15$ students were playing the game.