Question 1:

Let be real numbers different from 1, such that

and

also.

Then, what is the value of

?

Solution 1:

Note that

which is required answer.

Note that the maximum index 2014 plays no significant role here.

Question 2:

Let f be a one-to-one function from the set of natural numbers to itself such that

for all natural numbers m and n.

What is the least possible value of ?

Answer 2:

From elementary number theory, we know that given f is a multiplicative function and hence, the required function is such that if p and q are prime, then

That is we need to decompose 999 into its unique prime factorization.

So, we have where both 3 and 97 are prime.

We have and we want this to be least positive integer. Clearly, then f(3) cannot be greater than 97. Also, moreover, we need both f(3) and f(97) to be as least natural number as possible. So, and so that required answer is 24.

Question 3:

HW :

What is the number of ordered pairs (A,B) where A and B are subsets of such that neither nor ?

Cheers,

Nalin Pithwa

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