Problems
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Prove that if \begin{eqnarray} a_1&\leq& b_1\leq c_1,\\ a_2&\leq& b_2\leq c_2,\\ &\vdots&\\ a_N&\leq& b_N\leq c_N, \end{eqnarray} then \[ \sum_{i=1}^Na_i\leq \sum_{i=1}^Nb_i\leq \sum_{i=1}^Nc_i. \]
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Prove that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$, then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$.
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Prove that if $a_1\leq b_1$ and $a_2\leq b_2$, then $a_1+a_2\leq b_1+b_2$.
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Prove that if $a=bc$, $b\gt 0$ and $c\gt 1$, then $a\gt b$.
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Prove that if $a\gt 0$ and $b\gt 0$, then $a+b\gt a$ and $a+b\gt b$.
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Prove that if $0\lt a\lt b$, then $a^2\lt \frac{1}{3}\left(a^2+ab+b^2\right)\lt b^2$.
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Prove that if $0\lt x_1\lt x_2$, then $x^2_1\lt x^2_2$.
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Solve for $x$ in $5-3x\leq 8+5x$.
