Proof Writing

Let $x$ be a positive real number. Prove that $x(1-x)>0$ if $0<x<1$.

Prove that $m+n$ is even if the integers $m$ and $n$ have the same parity (i.e., both odd or both even).

Suppose that $a$ and $b$ are integers. Prove that if $a(b^{2}+5)$ is even then either $a$ is even or $b$ is odd.

Prove that no integers $a$ and $b$ exist such that $2a+4b=1$.

Prove that $\frac{a+b}{2}\geq\sqrt{ab}$ if $a$ and $b$ are positive real numbers by way of contradiction.

Prove using mathematical induction that $a^{2}-1$ is a multiple of 8 for all positive odd integers $a$.

Prove that every integer $n\geq 2$ is either prime or can be expressed as a product of primes.

Let $n$ be an integer. Prove that $n$ is one more than a multiple of 5 if and only if $n$ is of the form $5k-4$ where $k$ is an integer.

Prove that there is a positive integer that is one less than a perfect cube and two less than a perfect square.

Let $x=\sqrt{2}$ and $y=2\log_{2}{3}$. It may be assumed that $\sqrt{2}$ is irrational.

1. Prove that $y$ is irrational.

2. Produce a constructive proof that there exist irrational $x$ and $y$ such that $x^{y}$ is rational.