# Tagged Questions

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### An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
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### Solving inequalities with absolute values

This is the question: $$\left| \frac{x+2}{3(x-1)} \right| \leq \frac{2}{3}$$ And this is my working out, first I squared both the numerator and denominator, then solved it as if it was a normal ...
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### Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
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### Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
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### inequality fraction

I saw this problem in an Australian maths olympiad: $6/10 < a/b < 10/15$ The problem asked for the lowest possible value of $b\in \mathbb{Z}$. I tried manipulating but couldnt derive one of ...
I found this interesting problem in a programming forum. It would be great to get some help.. Solve for $K$, $K-1 \leq 27\cdot\log_{10}(9(K-1))$ We plotted it to find that $K\leq77$. Can you solve ...