2020/025) How many positive perfect squares less than $10^6$ are multiples of 24?

 Because the number is a perfect square and multiple of 24 this should be multiple of smallest multiple of 24 which is a square.

Now  $24 = 2^3 * 3 $ so to get the smallest multiple of 24 which is a square we need to make the power of 2 and 3 both even or we need to multiply by 2* 3 or 6 to get 144. So we need

$144n^2 = (12n)^2 <= 10^6$ 

or $12n <= 1000$

or $n <= \frac{1000}{12}= 83.3$

So $n < =83$ and hence there are 83 numbers

2020/024) Show that if for sides a,b,c of a triangle if for each integer n the sides $a^n,b^n,c^n$ form a triangle then the triangle is isosceles.

Without loss of generality let us assume that a is the longest side. if the triangle is not  isosceles then we have $ a > b$ and $a > c$.

For it to an triangle we need to have $a^n < b^n + c^n$

Or $$(\frac{b}{a})^n +  (\frac{c}{a})^n > 1\cdots(1)$$ for all n

As $b < a$ so $(\frac{b}{a}) < 1$ and  $(\frac{b}{a})^n < 1$ and as n goes to infinity this goes to zero. 

Similarly $(\frac{c}{a})^n < 1$ and as n goes to infinity this goes to zero. 

So the sum goes to zero and hence (1) is not true so $a^n,b^n.c^n$ sides cannot form a triangle

So  either b or c has to be same as a. So the triangle is isosceles.