A rigorous mathematical proof
Hi, these questions require a rigorous mathematical proof, based on the topic areas provided for each question. There must not be only an answer provided for each question, but a rigorous mathematical proof, followed by an explanation of how the proof works and how you came up with it please? Thank you.
Topic 1: Use the Pigeonhole Principle to solve the following
Q1: Suppose that if we have a 3 × 7 grid of squares, where each unit square is coloured either black or white. Prove that there exists a rectangle made up of the unit squares whose corner squares are of the same colour.
Q2: A disk of radius 1 is completely covered by 7 identical smaller disks (which may overlap). Show that the radius of the smaller disks must be at least 1/2.
Q3: Each of 14 red balls and 14 green balls is marked with an integer between 1 and 100 inclusive; no integer appears on more than one ball. The value of a pair of balls is the sum of the numbers on the balls. Show there are at least two pairs, consisting of one red and one green ball, with the same value.
Q4: Among a group of n integer numbers, prove that there is some subset of them whose sum is divisible by n.
Topic 2: Use divisibility theorems to solve these problems, by a rigorous proof.
Q1: Prove that every prime number greater than 3 leaves a remainder of either 1 or 5 when divided by 6.
Q2: If 935712 × N is a perfect cube for some positive integer N , find the minimum possible value for N.
Q3: Let p, q, r be prime numbers such that pqr (p + q + r) is a perfect square. Find all possible values of
(p, q, r).
Q4:
(a) What is the maximum number of terms in an arithmetic sequence of primes with common difference 6?
(b) What is the minimum common difference for an increasing arithmetic sequence of 6 primes.