MathDB

2

Part of 1989 China Team Selection Test

Problems(2)

There no terms with 3^x*5^y

Source: China TST 1989, problem 2

6/27/2005
Let v0=0,v1=1v_0 = 0, v_1 = 1 and vn+1=8⋅vn−vn−1,v_{n+1} = 8 \cdot v_n - v_{n-1}, n=1,2,...n = 1,2, .... Prove that in the sequence {vn}\{v_n\} there aren't terms of the form 3α⋅5β3^{\alpha} \cdot 5^{\beta} with α,β∈N.\alpha, \beta \in \mathbb{N}.
LaTeXnumber theory unsolvednumber theory
find the range of angle BAC

Source: China TST 1989, problem 6; China North MO

6/27/2005
ADAD is the altitude on side BCBC of triangle ABCABC. If BC+AD−AB−AC=0BC+AD-AB-AC = 0, find the range of ∠BAC\angle BAC. Alternative formulation. Let ADAD be the altitude of triangle ABCABC to the side BCBC. If BC+AD=AB+ACBC+AD=AB+AC, then find the range of ∠A\angle{A}.
trigonometrygeometryfunctionangle bisectorgeometry solved