MathDB

2

Part of 2011 Brazil Team Selection Test

Problems(2)

n not prime if a^{n-1}=1 mod n, a^{{n-1}/q} not 1 mod n

Source: 2011 Brazil IMO TST 3.2

7/24/2021
Let n≥3n\ge 3 be an integer such that for every prime factor qq of n−1n-1 exists an integer a>1a > 1 such that an−1≡1 (mod  n )a^{n-1} \equiv 1 \,(\mod n \, ) and an−1q≢1 (mod  n )a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, ). Prove that nn is not prime.
number theoryprime
MS = MT iff X,Y,Z,W are concyclic, starting with 2 intersecting circles

Source: 2011 Brasil IMO TST 4.2

7/23/2021
Given two circles ω1\omega_1 and ω2\omega_2, with centers O1O_1 and O2O_2, respectively intesrecting at two points AA and BB. Let XX and YY be points on ω1\omega_1. The lines XAXA and YAYA intersect ω2\omega_2 again in ZZ and WW, respectively, such that AA is between X,ZX,Z and AA is between Y,WY,W. Let MM be the midpoint of O1O2O_1O_2, S be the midpoint of XAXA and TT be the midpoint of WAWA. Prove that MS=MTMS = MT if, and only if, the points X,Y,ZX, Y, Z and WW are concyclic.
Concyclicgeometryequal segments