MathDB

3

Part of 2018 Thailand TSTST

Problems(3)

TR = TS for intersecting circles

Source: 2017 Thailand October Camp 1.3

10/15/2020
Circles O1,O2O_1, O_2 intersects at A,BA, B. The circumcircle of O1BO2O_1BO_2 intersects O1,O2O_1, O_2 and line ABAB at R,S,TR, S, T respectively. Prove that TR=TSTR = TS
geometrycircumcircle
fixed point and perpendicular wanted, 2 circles, tangents

Source: 2017 Thailand October Camp 2.3

10/15/2020
Let BCBC be a chord not passing through the center of a circle ω\omega. Point AA varies on the major arc BCBC. Let EE and FF be the projection of BB onto ACAC, and of CC onto ABAB respectively. The tangents to the circumcircle of △AEF\vartriangle AEF at E,FE, F intersect at PP. (a) Prove that PP is independent of the choice of AA. (b) Let HH be the orthocenter of △ABC\vartriangle ABC, and let TT be the intersection of EFEF and BCBC. Prove that TH⊥APTH \perp AP.
geometryfixedperpendicularFixed point
n|1+m^(3^n)+m^(2*3^n)

Source: 2017 Thailand October Camp 3.3

2/25/2022
Find all pairs of integers m,n≥2m, n \geq 2 such that n∣1+m3n+m2⋅3n.n\mid 1+m^{3^n}+m^{2\cdot 3^n}.
number theoryDivisibilityBulgaria