MathDB

2

Part of 2001 China Team Selection Test

Problems(2)

China TST 2001 natural number sequence

Source: China TST 2001, problem 2

5/22/2005
aa and bb are natural numbers such that b>a>1b > a > 1, and aa does not divide bb. The sequence of natural numbers {bn}n=1∞\{b_n\}_{n=1}^\infty satisfies bn+1≥2bn∀n∈Nb_{n + 1} \geq 2b_n \forall n \in \mathbb{N}. Does there exist a sequence {an}n=1∞\{a_n\}_{n=1}^\infty of natural numbers such that for all n∈Nn \in \mathbb{N}, an+1−an∈{a,b}a_{n + 1} - a_n \in \{a, b\}, and for all m,l∈Nm, l \in \mathbb{N} (mm may be equal to ll), am+al∉{bn}n=1∞a_m + a_l \not\in \{b_n\}_{n=1}^\infty?
algebra unsolvedalgebracombinatoricsnumber theory
China TST 2001 circumcenter and incenter

Source: China TST 2001, problem 5

5/22/2005
In the equilateral △ABC\bigtriangleup ABC, DD is a point on side BCBC. O1O_1 and I1I_1 are the circumcenter and incenter of △ABD\bigtriangleup ABD respectively, and O2O_2 and I2I_2 are the circumcenter and incenter of △ADC\bigtriangleup ADC respectively. O1I1O_1I_1 intersects O2I2O_2I_2 at PP. Find the locus of point PP as DD moves along BCBC.
geometrycircumcircleincentergeometry solved