MathDB

1

Part of 2006 Turkey Team Selection Test

Problems(2)

Maximum area of a heptagon

Source: Turkey, TST D1, P1

5/10/2006
Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.
geometrycircumcirclequadraticstrigonometryPythagorean Theoremalgebrageometry proposed
Find least integer such that 2006 divides x_{2006}

Source: Turkey, TST D2, P1

5/10/2006
For all integers n≥1n\geq 1 we define xn+1=x12+x22+⋯+xn2x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2, where x1x_1 is a positive integer. Find the least x1x_1 such that 2006 divides x2006x_{2006}.
inductionquadraticsGaussmodular arithmeticnumber theoryprime numbersnumber theory proposed