91Ó°ÊÓ

Show that postage of 24 cents or more can be achieved by using only 5 -cent and 7 -cent stamps.

Prove that for all \(x \in \mathbf{R}\), if \(x^{3}\) is irrational, then \(x\) is irrational.

Prove that \(\sqrt[3]{2}\) is irrational.

Prove that for all integers \(m\) and \(n\), if \(m\) and \(n\) are odd, then \(m n\) is odd.

Using induction, verify that each equation is true for every positive integer \(n\). $$ \begin{array}{l} \cos x+\cos 2 x+\cdots+\cos n x=\frac{\cos [(x / 2)(n+1)] \sin (n x / 2)}{\sin (x / 2)} \\ \text { provided that } \sin (x / 2) \neq 0 \end{array} $$

Using induction, verify the inequality. $$ \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots(2 n)} \leq \frac{1}{\sqrt{n+1}}, n=1,2, \ldots $$

Using induction, verify the inequality. $$ 2^{n} \geq n^{2}, n=4,5, \ldots $$

Suppose that we have two piles of cards each containing \(n\) cards. Two players play a game as follows. Each player, in turn, chooses one pile and then removes any number of cards, but at least one, from the chosen pile. The player who removes the last card wins the game. Show that the second player can always win the game.

Using induction, verify the inequality. Use the geometric sum to prove that $$ r^{0}+r^{1}+\cdots+r^{n}<\frac{1}{1-r} $$ for all \(n \geq 0\) and \(0

Prove or disprove: \((X-Y) \cap(Y-X)=\varnothing\) for all sets \(X\) and \(Y\).