Mathematics Hothouse

Problem 1:

If a, b, c, and d satisfy the equations

then what is the numerical value of ?

Problem 2:

Suppose x and y are positive integers with and and when divided by 5, leave remainders 2 and 3, respectively. It follows that when is divided by 5, the remainder is necessarily equal to

(A) 2

(B) 1

(C) 4

(D) none of the foregoing numbers

Problem 3:

The number of different solutions of  the equation , where each of x, y, and z is a positive integer is

(A) 36

(B) 121

(C)

(D) , which denote binomial coefficients

Problem 4:

The hands of a clock are observed simultaneously from 12.45 pm onwards. They will be observed to point in the same direction some time between

(A) 1:03 pm and 1:04 pm

(B) 1:04 pm and 1:05pm

(C) 1:05 pm and 1:06 pm

(D) 1:06 pm and 1:07 pm.

More later,

Nalin Pithwa