A positive integer is called vaivém when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, 2021 is vaivém, as 2 > 0 and 0 < 2 and 2 > 1. The number 2023 is not vaivém, as 2 > 0 and 0 < 2, but 2 is not greater than 3.

a) How many vaivém positive integers are there from 2000 to 2100?

b) What is the largest vaivém number without repeating digits?

c) How many distinct 7-digit numbers formed by all the digits 1, 2, 3, 4, 5, 6 and 7 are vaivém?

If x and y are real numbers, then solve this system of equations:

√x + √y =3,

√(x+5) + √(y+3) = 5

A 11-digit number is such that it contains each of the digits from 1 to 9

*at least once*.

What is the percentage of prime numbers in such an occurrence?

Show that for any prime p there exists a nonnegative integer N such that:

2^N+3^N+6^N-1 is a multiple of p.