XXIX OM - II - Problem 4

From the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\lim_{n\to \infty} p_n$.
Note. We assume that all choices of three different points are equally probable.
Let $ A_1, A_2, \ldots , A_{2n} $ be the consecutive vertices of a regular $ 2n $-gon, and $ O $ - the center of the circle circumscribed around this polygon. We will investigate how many triangles $ A_1A_iA_j $, where $ i < j $, have the angle $ \measuredangle A_iA_1A_j $ not acute.
Since the inscribed angle in a circle has a measure that is half the measure of the central angle subtended by the same arc, therefore

Thus, $ \measuredangle A_iA_1A_j $ is not acute if and only if $ j - i \geq n $. We will calculate the number of such pairs $ (i, j) $, such that

From these inequalities, it follows that $ i \leq j - n \leq 2n - n = n $. For each $ i $ satisfying $ 2 \leq i \leq n $, there are $ n - i + 1 $ values for $ j $ satisfying (1), namely $ j = i + n, i + n + 1, \ldots, 2n $. Therefore, the number of such pairs is

We have thus proved that the number of non-acute triangles $ A_1A_iA_j $, in which the angle $ \measuredangle A_iA_1A_j $ is not acute, is equal to $ \frac{n(n-1)}{2} $. Since in a triangle at most one angle is not acute, the number of all non-acute triangles $ A_iA_jA_k $ is equal to

The number of all triangles $ A_iA_jA_k $ is equal to the number of three-element subsets of the $ 2n $-element set $ \{A_1, A_2, \ldots , A_{2n} \} $, that is, the number

Therefore,

Thus, $ \displaystyle \lim_{n \to \infty} p_n = 1 - \frac{3}{4} = \frac{1}{4} $.
om29_2r_img_12.jpg
Note. The triangle $ A_1A_iA_j $, where $ 1 < i < j \leq 2n $, is acute if and only if $ i \leq n $, $ j > n $, and $ j - i < n $. By associating the triangle $ A_1A_iA_j $ with the point $ (i,j) $, we see that the acute triangles correspond to lattice points located in the shaded triangle (Fig. 12), and any triangles correspond to lattice points belonging to the triangle $ AOB $. The ratio of the number of lattice points in the shaded triangle to the number of all lattice points in the triangle $ AOB $ approaches the ratio of the areas of these triangles as $ n \to \infty $. This ratio of areas is equal to $ \frac{1}{4} $.