MathDB

Problem 1

Part of 2009 VJIMC

Problems(2)

incenter of AIB, etc., limit of sequence

Source: VJIMC 2009 1.1

6/12/2021
Let ABCABC be a non-degenerate triangle in the euclidean plane. Define a sequence (Cn)n=0∞(C_n)_{n=0}^\infty of points as follows: C0:=CC_0:=C, and Cn+1C_{n+1} is the incenter of the triangle ABCnABC_n. Find lim⁡n→∞Cn\lim_{n\to\infty}C_n.
geometryincenterlimits
self-descriptive number in base b

Source: VJIMC 2009 2.1

6/12/2021
A positive integer mm is called self-descriptive in base bb, where b≥2b\ge2 is an integer, if
i) The representation of mm in base bb is of the form (a0a1…ab−1)b(a_0a_1\ldots a_{b-1})_b (that is m=a0bb−1+a1bb−2+…+ab−2b+ab−1m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}, where 0≤ai≤b−10\le a_i\le b-1 are integers). ii) aia_i is equal to the number of occurences of the number ii in the sequence (a0a1…ab−1)(a_0a_1\ldots a_{b-1}).
For example, (1210)4(1210)_4 is self-descriptive in base 44, because it has four digits and contains one 00, two 11s, one 22 and no 33s.
Basesnumber theory