MathDB

2

Part of 2008 Argentina Iberoamerican TST

Problems(2)

Prime number

Source: CWMO 2005-3

11/25/2005
Set S={1,2,3,...,2005}S = \{1, 2, 3, ..., 2005\}. If among any nn pairwise coprime numbers in SS there exists at least a prime number, find the minimum of nn.
number theoryprime numbersrelatively primenumber theory solved
Prove TZ parallel to XY

Source: Argentina TST Iberoamerican 2008 problem 2

8/27/2009
Two circunmferences Γ1 \Gamma_1 Γ2 \Gamma_2 intersect at A A and B B r1 r_1 is the tangent from A A to Γ1 \Gamma_1 and r2 r_2 is the tangent from B B to Γ2 \Gamma_2 r_1 \cap r_2\equal{}C T\equal{} r_1 \cap \Gamma_2 (T≠A T \neq A) We consider a point X X in Γ1 \Gamma_1 which is distinct from A A and B B. XA \cap \Gamma_2 \equal{}Y (Y≠A Y \neq A) YB \cap XC\equal{}Z Prove that TZ∥XY TZ \parallel XY
geometry unsolvedgeometry