MathDB

2

Part of 1998 Vietnam Team Selection Test

Problems(2)

lots of tangent circles

Source: Vietnam TST 1998 for the 39th IMO, problem 2

6/26/2005
In the plane we are given the circles Γ\Gamma and Δ\Delta tangent to each other and Γ\Gamma contains Δ\Delta. The radius of Γ\Gamma is RR and of Δ\Delta is R2\frac{R}{2}. Prove that for each positive integer n≥3n \geq 3, the equation: (p(1)−p(n))2=(n−1)2⋅(2⋅(p(1)+p(n))−(n−1)2−8) (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) is the necessary and sufficient condition for nn to exist nn distinct circles Υ1,Υ2,…,Υn\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n such that all these circles are tangent to Γ\Gamma and Δ\Delta and Υi\Upsilon_i is tangent to Υi+1\Upsilon_{i+1}, and Υ1\Upsilon_1 has radius Rp(1)\frac{R}{p(1)} and Υn\Upsilon_n has radius Rp(n)\frac{R}{p(n)}.
Pythagorean Theoremgeometrygeometry unsolved
let d be a positive divisor of 5 + 1998^{1998}

Source: Vietnam TST 1998 for the 39th IMO, problem 5

6/26/2005
Let dd be a positive divisor of 5+199819985 + 1998^{1998}. Prove that d=2⋅x2+2⋅x⋅y+3⋅y2d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2, where x,yx, y are integers if and only if dd is congruent to 3 or 7 (mod20)\pmod{20}.
modular arithmeticquadraticsnumber theory unsolvednumber theory