how many numbers have this property: when you subtract 297 from it, you get the same number but in reverse.

9514 1404 393

  60

Step-by-step explanation:

Consider the number with digits x, y, z. Its value will be 100z+10y+z. We want the same value when we reverse the digits after subtracting 297.

  100x +10y +z -297 = 100z +10y +x

  99x -99z = 297 . . . . . subtract 100z+10y+x-297

  x -z = 3 . . . . . . . . . . . . divide by 99

The first digit of the number is 3 more than the last digit. There are 60 such numbers. The first digit may be 4..9, and the middle digit may be 0..9.

  401, 502, 603, ..., 906; 411, 512, 613, ..., 916, ..., 491, 592, 693, ... 996

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

c > -10.55

step-by-step explanation: