What is the radius of the table?

[loli12]

Anyone has any clue to these questions? They are from a contest last year.
Please Help and if you don't mind... please explain in detail.. Thanks!

1. A library has between 1000 and 2000 books. Of these, 25% are fiction, 1/13 are biographies and 1/17 are atlases. How many books are either biographies or atlases?

2. A circular table is pushed into a corner of a rectangular room so that it touches both walls. A point on the edge of the table between the two points of contact is 2 inches from one wall and 9 inches from the other wall. What is the radius of the table?

3. Let f(x) = ax + b, with b<a both positive integers. If for positive integers p and q, f(p) = 18 and f(q) = 39, what is the value of b?

4. In Triangle SBC, SB = 12, BC = 15, and SC=18. Let O be the point for which BO bisects angle SBC and CO bisects angle SCB. IF M and L are on sides SB and SC respectively so that ML is parallel to side BC and contains point O, what is the perimeter of Triangle SML?

 

[Gokul43201]

1. What number between 1000 and 2000 can be divided exactly by 4, 13 and 17 ?

2. Let the co-ordinates of the point be (rcos\theta, rsin\theta ). You now have 2 equations in 2 unknowns.

3. Subtract the two equations. You get only 4 possible choices for the values of a. Three of these are ruled out by inspection - in fact, it's plain to see why. The surviving one works.

4. Let SO extended meet BC at R. Since O is the incenter, we have (I'm not sure what this rule is called) SO/OR = (12+18)/15. From here, it's just similar triangles.