MathDB

1

Part of 2000 IMO Shortlist

Problems(2)

Shortlist 2000

Source: IMO Shortlist 2000, Problem G1

8/16/2003
In the plane we are given two circles intersecting at X X and Y Y. Prove that there exist four points with the following property: (P) For every circle touching the two given circles at A A and B B, and meeting the line XY XY at C C and D D, each of the lines AC AC, AD AD, BC BC, BD BD passes through one of these points.
geometrytrapezoidcombinatorial geometrycirclesIMO Shortlist
Very easy number theory

Source: IMO Shortlist 2000, N1, 6th Kolmogorov Cup, 1-8 December 2002, 1st round, 1st league,

8/6/2004
Determine all positive integers n≥2 n\geq 2 that satisfy the following condition: for all a a and b b relatively prime to n n we have a≡b(modn)if and only ifab≡1(modn).a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.
modular arithmeticEulernumber theoryIMO Shortlistnumber theory solved