MathDB

2

Part of 2005 Vietnam National Olympiad

Problems(2)

x!+y!/n!=3^n

Source: VietNamMO 2005,problem5

3/11/2005
Find all triples of natural (x,y,n) (x,y,n) satisfying the condition: \frac {x! \plus{} y!}{n!} \equal{} 3^n Define 0! \equal{} 1
Ross Mathematics Programnumber theory proposednumber theory
A geometry problem from VMO 2005

Source: Vietnam MO 2005,problem2

3/10/2005
Let (O)(O) be a fixed circle with the radius RR. Let AA and BB be fixed points in (O)(O) such that A,B,OA,B,O are not collinear. Consider a variable point CC lying on (O)(O) (C≠A,BC\neq A,B). Construct two circles (O1),(O2)(O_1),(O_2) passing through A,BA,B and tangent to BC,ACBC,AC at CC, respectively. The circle (O1)(O_1) intersects the circle (O2)(O_2) in DD (D≠CD\neq C). Prove that: a) CD≤R CD\leq R b) The line CDCD passes through a point independent of CC (i.e. there exists a fixed point on the line CDCD when CC lies on (O)(O)).
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