MathDB

2

Part of 2015 Iran Team Selection Test

Problems(3)

Easy Geometry

Source: Iran TST 2015, exam 1, day 1 problem 2

5/10/2015
IbI_b is the BB-excenter of the triangle ABCABC and ω\omega is the circumcircle of this triangle. MM is the middle of arc BCBC of ω\omega which doesn't contain AA. MIbMI_b meets ω\omega at T≠MT\not =M. Prove that TB⋅TC=TIb2. TB\cdot TC=TI_b^2.
geometry
number theory

Source: iranian TST 2015 third exam day 1 P2

6/12/2015
Assume that a1,a2,a3a_1, a_2, a_3 are three given positive integers consider the following sequence: an+1=lcm[an,an−1]−lcm[an−1,an−2]a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}] for n≥3n\ge 3 Prove that there exist a positive integer kk such that k≤a3+4k\le a_3+4 and ak≤0a_k\le 0. ([a,b][a, b] means the least positive integer such thata∣[a,b],b∣[a,b] a\mid[a,b], b\mid[a, b] also because lcm[a,b]\text{lcm}[a, b] takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
number theoryIranIranian TSTleast common multiple
easy geometry

Source: iranian TST second exam p2

6/3/2015
In triangle ABCABC(with incenter II) let the line parallel to BCBC from AA intersect circumcircle of △ABC\triangle ABC at A1A_1 let AI∩BC=DAI\cap BC=D and EE is tangency point of incircle with BCBC let EA1∩⊙(△ADE)=T EA_1\cap \odot (\triangle ADE)=T prove that AI=TIAI=TI.
geometryincentercircumcircle