MathDB

Problem 6

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

side ratio in triangle, angle bisectors form cyclic quadrilateral

Source: Mongolia MO 2000 Grade 10 P6

4/22/2021
In a triangle ABCABC, the angle bisector at A,B,CA,B,C meet the opposite sides at A1,B1,C1A_1,B_1,C_1, respectively. Prove that if the quadrilateral BA1B1C1BA_1B_1C_1 is cyclic, then ACAB+BC=ABAC+CB+BCBA+AC.\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.
geometryangle bisectorratiocyclic quadrilateral
# of naturals not exceeding n divisible by exactly one of a set of primes

Source: Mongolia MO 2000 Teachers P6

4/22/2021
Given distinct prime numbers p1,…,psp_1,\ldots,p_s and a positive integer nn, find the number of positive integers not exceeding nn that are divisible by exactly one of the pip_i.
number theory