1) Find the remainder of the division of \( 3^{100} \) by 100.

2) Let \( a, b, c \) be three strictly positive real numbers such that \( a + b + c = 1 \). What is the value of \( \frac{1}{a^2+1} + \frac{1}{b^2+1} + \frac{1}{c^2+1} \)?

3) What is the sum of all positive integers such that \( (n-1)! + n! = (n+1)! \)?

4) Let \( a, b, c \) be three integers such that \( a^2 + b^2 = c^2 \). Suppose that \( c + b < 50 \) and \( a < 10 \). What is the maximum sum of \( a + b + c \)?

5) Find the sum of the roots of \( x^4 - 8x^3 + 18x^2 - 8x = 0 \).

6) Let \( f \) be a function from \( \mathbb{R} \) to \( \mathbb{R} \) such that \( f(x) = x^3 - 3x^2 - 9x - 5 \). Calculate \( f(f(f(2))) \).

7) Find an integer \( 100 < n < 2025 \) such that the sum of the digits of \( n \) and \( 2n \) is equal to 1234.

8) Find the number of integers \( n \) such that \( 2^{n-1} \cdot n + 1 \) is a perfect square.

9) Find the number of integers \( n \) such that \( 2^n + 17 \) is a perfect square.

10) What is the sum of all prime numbers \( p \) such that $$ 16p^{2024}(p^{2023}+2025)$$ is a perfect square and is divisible by 6.?

11) a,b and c are 3 positif reel numbers such that abc=1.Suppose that c>0 find the minimum of the expression $$a^{3}+b^{3}+c^{3}$$.

12) Find the largest number of consecutive integers starting from 1 that can be the sum of the squares of the digits of some integer.

13) If \(a, b, c\), and \(d\) are 4 integers such that \(a + b + c + d = 2025\) and \( (abcd)^2 = 2025 \), find the sum of \(a^4 + b + d^2 + 7\).