1. The tangents of the angles of a triangle are natural numbers. What can they be equal to?
1. The first method. Let the angles of the triangle be denoted by $\alpha, \beta, \gamma (\alpha \leqslant \beta \leqslant \gamma)$. Since $\alpha+\beta+\gamma=\pi$, the inequality $02, n>3$. In this case, it would be necessary to increase two angles of the triangle $ABC$, leaving the third unchanged, which is impossible by the same theorem about the sum of the angles of a triangle. Thus, the variant in the figure is the only one.