MIND BLOWING 1

1. Find the side length of the smallest equilateral triangle in which
three discs of radii 2, 3, 4 can be placed without overlap.

2. The quadrilateral ABCD is inscribed in a circle. The lines AB
and CD meet at E, while the diagonals AC and BD meet at F.
The circum circles of the triangles AFD and BFC meet again at H.
Find angle (EHF)?

3. A 7 x 7 chessboard is given with its four corners deleted.
(a) What is the smallest number of squares which can be colored
black so that an uncolored 5-square (Greek) cross cannot be
found?
(b) Prove that an integer can be written in each square such that
the sum of the integers in each 5-square cross is negative while
the sum of the numbers in all squares of the board is positive.

4. Starting at (1; 1), a stone is moved in the coordinate plane according
to the following rules:
(i) From any point (a; b), the stone can move to (2a; b) or (a; 2b).
(ii) From any point (a; b), the stone can move to (a - b; b) if a > b,
or to (a; b - a) if a < b.
For which positive integers x; y can the stone be moved to (x; y)?

5. Each diagonal of a convex pentagon is parallel to one side of the
pentagon. Prove that the ratio of the length of a diagonal to that of
its corresponding side is the same for all five diagonals, and compute
this ratio.

6. For each positive integer n, and the greatest common divisor of n!+1
and (n + 1)!.

7. Let A and B be opposite vertices of a cube of edge length 1. Find
the radius of the sphere with center interior to the cube, tangent to
the three faces meeting at A and tangent to the three edges meeting
at B.

8. A function f is de fined on the positive integers satis fies f(1) = 2010
and
f(1) + f(2) + ............ + f(n) = (n^2)f(n) (n > 1):
Calculate f(2010).

9. Let n be a natural number. A cube of side length n can be divided
into 1996 cubes whose side lengths are also natural numbers.
Determine the smallest possible value of n.

10. The numbers from 1 to 37 are written in a line so that each number
divides the sum of the previous numbers. If the first number is 37
and the second number is 1, what is the third number?