\documentclass[12pt]{article}
\usepackage{tipx}
\usepackage{textcomp}
\begin{document}
\title {Example Here}
\author{Example Here}
\date {Due October 12, 2016}
\maketitle
\textbf{Exercise 1} \\
Show that this implication is a tautology, by using a truth table: \\
\[ [(P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)] \rightarrow R \] \\
\pagebreak
\textbf{Exercise 2} \\
Show that the following is a tautology: \\
\[ (P \vee Q) \wedge (\neg{P} \vee R ) \rightarrow (Q \vee R) \] \\
\pagebreak
\textbf {Exercise 3} \\
a) Let x be a real number. Show that is \[x^2\] is irrational, then x is irrational.\\
b) Based on question a), can you say that "if x is irrational, it follows that \[x^2\] is irrational." ? \\
\textbf {Exercise 4} \\
Prove that a square of an integer ends with a 0, 1, 4, 5, 6, or 9.
(Hint: let n = 10k + l, where l = 0, 1, ... , 9) \\
\pagebreak
\textbf {Exercise 5} \\
Prove that if n is a positive integer, then n is even if and only if 5n + 6 is even.\\
\pagebreak
\textbf {Exercise 6} \\
Prove that either \[3.10^450 + 15\] or \[3.10^450 + 16\] is not a perfect square.
Is your proof constructive, or non-constructive? \\
\pagebreak
\textbf {Exercise 7} \\
Prove or disprove that if a and b are rational numbers, then \[a^b\] is also rational.\\
\textbf {Exercise 8} \\
Prove that at least one of the real numbers \[a_1, a_2, ... , a_n\] is greater than or equal to the average of these numbers. What kind of proof did you use?\\
\textbf {Exercise 9} \\
The proof below has been scrambled. Please put it back in the correct order.\\
\end{document}
我的新问题是，如何才能获得 x^2 而不让它跳转到新行？我希望它留在句子中。我试过 Ensuremath，但无论我做什么都不起作用。请测试代码并亲自查看。
答案1
发布的文档显示
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由于 10000 是 TeX 报告的最差结果，因此日志应被视为唯一的结果，我没有查看 pdf，
修复一些简单的问题，删除所有\\，使用``和''不"使用\item来枚举项目，使用$内联数学，使用{}来分隔上标，并使用数学来表示所有数学实例而不仅仅是你需要的地方，^所以字体是一致的，删除所有警告并产生
\documentclass[12pt]{article}
\usepackage{enumitem}
\begin{document}
\title {Example Here}
\author{Example Here}
\date {Due October 12, 2016}
\maketitle
\begin{enumerate}[label=\textbf{Example \arabic*}]
\item
Show that this implication is a tautology, by using a truth table:
\[ [(P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)] \rightarrow R \]
\item
Show that the following is a tautology:
\[ (P \vee Q) \wedge (\neg{P} \vee R ) \rightarrow (Q \vee R) \]
\item
\begin{enumerate}[label=\alph*)]
\item Let $x$ be a real number. Show that is $x^2$ is irrational, then $x$ is irrational.
\item Based on question a), can you say that ``if $x$ is irrational, it follows that $x^2$ is irrational.''?
\end{enumerate}
\item
Prove that a square of an integer ends with a 0, 1, 4, 5, 6, or 9.
(Hint: let n = 10k + l, where l = 0, 1, ... , 9)
\item
Prove that if $n$ is a positive integer, then $n$ is even if and only if $5n + 6$ is even.
\item
Prove that either $3.10^{450} + 15$ or $3.10^{450} + 16$ is not a perfect square.
Is your proof constructive, or non-constructive?
\item
Prove or disprove that if $a$ and $b$ are rational numbers, then $a^b$ is also rational.
\item
Prove that at least one of the real numbers $a_1, a_2, \ldots , a_n$ is greater than or equal to the average of these numbers. What kind of proof did you use?
\item
The proof below has been scrambled. Please put it back in the correct order.
\end{enumerate}
\end{document}