**Problem 1.**

Determine all positive integers such that,

for some , where is the number of divisors of .

Determine all positive integers such that,

for some , where is the number of divisors of .

Find the first digit before and after the decimal point in . (without using a calculator)

points are chosen on a circle. In how many ways can you join pairs of points by nonintersecting chords?

Let be a function, continuous on [a,b] and differentiable on (a,b). Prove that if there exists such that

then there exists such that

Answer for problem 4:

case 1:: f(b)<f(c) and f(a)f(b)-f(c)f'(j)=[f(b)-f(c)]+[f(c)-f(a)]/ (b-a)

Clearly if l f(b)-f(c) l 0 else f'(j)<0

Case 2:: f(c)f(c)

By similar way we can say that there exist k belongs to (a, b) such that f'(k)<0 when l f(b)-f(c) l 0.

Combining cases we say that there exist two points j and k such that f'(j)f'(k) there exist i in (a, b) such that f'(i)=0.

Note :::if l f(b)-f(c) l = l f(c)-f(a) l then f'(j)=f'(k)=0.

This complete the proof.