## Pre-RMO — training

Problem 1:

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

$a+7b+3c+5d=0$

$8a+4b+6c+2d=-16$

$2a+6b+4c+8d=16$

$5a+3b+7c+d=-16$

then what is the numerical value of $(a+d)(b+c)$?

Problem 2:

Suppose x and y are positive integers with $x>y$ and $3x+2y$ and $2x+3y$ when divided by 5, leave remainders 2 and 3, respectively. It follows that when $x-y$ 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 $(x,y,z)$ of  the equation $x+y+z=10$, where each of x, y, and z is a positive integer is

(A) 36

(B) 121

(C) $10^{3}-10$

(D) $C_{3}^{10}-C_{2}^{10}$, 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

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