3. Find all pairs of numbers (a, b) for which the equality $(\mathrm{a}+\mathrm{b}-1)^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}-1$ holds.
Answer: all pairs of numbers $(1, t),(t, 1)$, where $t$ is any number.
By moving all terms to one side and using the difference of squares formula and factoring out the common factor, we get $2(a-1)(b-1)=0$. If the product is zero, then one of the factors is zero. Hence the answer.

Criteria. It is obtained that only one of the variables is equal to 1 due to division by an expression that may be equal to zero: 2 points.