MathDB

3

Part of 2005 May Olympiad

Problems(2)

Angles of the triangle

Source: May Olympiad(Olimpiada de Mayo) 2005

2/21/2018
In a triangle ABCABC with AB=ACAB = AC, let MM be the midpoint of CBCB and let DD be a point in BCBC such that ∠BAD=∠BAC6\angle BAD = \frac{\angle BAC}{6}. The perpendicular line to ADAD by CC intersects ADAD in NN where DN=DMDN = DM. Find the angles of the triangle BACBAC.
geometry
assign 0 and 1 at 99 collinear points

Source: XI May Olympiad (Olimpiada de Mayo) 2005 L1 P3

9/22/2022
A segment ABAB of length 100100 is divided into 100100 little segments of length 11 by 9999 intermediate points. Endpoint AA is assigned 00 and endpoint BB is assigned 11. Gustavo assigns each of the 9999 intermediate points a 00 or a 11, at his choice, and then color each segment of length 11 blue or red, respecting the following rule: The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. Determine if Gustavo can assign the 00's and 11's so as to get exactly 3030 blue segments. And 3535 blue segments? (In each case, if the answer is yes, show a distribution of 00's and 11's, and if the answer is no, explain why).
combinatorics