Part 2 of 3 — solutions to Pre RMO Oct 2014

Question Paper Set A:

11) For natural numbers x and y, let $(x,y)$ denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with $x \leq y$ satisfy the equation $xy= x+y+ (x,y)$ ?

Solution:

Here, if the condition, $x \leq y$ were not there, the answer would be infinitely many. But, so, just substitute some small numbers and check what is happening. In fact, in any number theory problem, first we should play with small numbers in our head.

Upon substitution, you will find that only $(2,3), (2,4), (3,3)$ satisfy the equation. Hence, the answer is 3.

12) Let ABCD be a convex quadrilateral with

$\angle DAB = \angle BDC = 90 \deg$ . Let the incircles of triangles ABD and BCD touch BD at P and Q respectively, with P lying in between B and Q. If

$AD=999$ and $PQ=200$ then what is the sum of the radii of the incircles of triangles ABD and BDC?

Solution: Please download the jpg fig attached.

The main properties to be used are Pythagoras’ theorem, that the angle bisectors of the vertices of a triangle meet at its incenter, the $\tan 2A$ formula of a triangle.

Let $C_{1}$ and $C_{2}$ be the incenter of $\Delta ABD$ and

$\Delta BCD$ respectively. Then, as shown in the attached figure:

Since, $\Delta APD$ is a right angled isosceles triangle, $DP=DQ=999$ , $\alpha 22.5 \deg$ , so

$\tan 45 \deg = \frac {\tan 22.5 \deg}{1- \tan^{2} 22.5 \deg}= r_{1}/999$ . Similarly, find the other radius and sum up the two.

Question 18.

Let f be a one-to-one function from the set of natural numbers to itself such that $f(mn)=f(m)f(n)$ for all natural numbers m and n. What is the least possible value of $f(999)$ ?

Probable Solution.

Use the Euler $\phi$ function. But, the answer this gives is different from the answer key.

Please send your comments, suggestions to this.

More later,

Nalin Pithwa

Mathematics Hothouse