Define the cost of a lattice point (i, j) to be i × j. A particle starts at (0, 0) and makes a series of 10 hops. Each hop increases the particle’s x-coordinate by 1 or y-coordinate by 1. Let the price of the particle’s path be the sum of the costs of the lattice points it meets, including (0, 0) and its final position. Find the maximum price of any path the particle can take.
Determine the volume generated by the revolution of the conic
x = a cosθ and y = b sinθ
about the line x = 2a.
If g is the inverse of f and f'(x)=1/(1+x^3), then prove that g'(x)=1+g'(x)^3.
Determine all possible values of the expression x - [x/2] - [x/3] - [x/6] by varying x in the real numbers.
If F(n) is a function that satisfies F(1)=F(2)=F(3)=1,with:
F(n+1)=(F(n)*F(n-1)+1)/F(n-2) whenever n>=3.
Find the value of F(6)
The point P lies on side CD of rectangle ABCD, with CD = 20 and AD = 10. The
circumcircle ω of △ABP re-intersects CD at Q. Given that the radius of ω is 11, find the distance
PQ.
Coins of diameter 1 have been placed in a square of side 11, without overlapping or protruding from the square. Can there be 126 coins? and 127? and 128?
