Let be an equilateral triangle of side length . Point is chosen on side such that the length of is . Likewise points and are chosen on sides and . with . Line segments are drawn from points to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled . Prove that is an equilateral triangle with side length .

Show that 3^(n+2) + 9^(n+1) + 4^(2n+1) + 4^(4n+1) is divisible by 13 for every n natural number.

How many 5 digit numbers are there such that sum of its digits is 18

Let . Show that

Let
then

Let . Prove that Let . Prove that

"Let N be a natural number. Show that any two numbers from the following sequence are coprime:

2^1 + 1, 2^2 + 1, 2^4+ 1,2^8+1 ..., 2^(2^N )+ 1."

Given such that .
Find the minimum value of .

Source: Pham Le Van

The polynomial is given by P(x) = x^4 + ax^3 + bx^2 + cx + d, and its roots are x1, x2, x3, x4. Additionally, it is stated that d ≥ 5.Find the minimum value of the product:

(x1^2 + 1)(x2^2 + 1)(x3^2 + 1)(x4^2 + 1).

Let and Prove that