query: Proof that $ k^2<2^k$
positive_passages.docid: 20964
positive_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 118467
negative_passages.text: Prove that $a^2+ab+b^2\ge 0$
negative_passages.docid: 62177
negative_passages.text: Prove that: $2^n < n!$ Using Induction
negative_passages.docid: 152104
negative_passages.text: How do I prove that $ij = k?$
negative_passages.docid: 44403
negative_passages.text: Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
negative_passages.docid: 153226
negative_passages.text: Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
negative_passages.docid: 2455
negative_passages.text: Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 27911
negative_passages.text: How can I prove that $xy\leq x^2+y^2$?
negative_passages.docid: 113168
negative_passages.text: If $a > b$, is $a^2 > b^2$?
negative_passages.docid: 92784
negative_passages.text: Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
negative_passages.docid: 13525
negative_passages.text: Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
negative_passages.docid: 4201
negative_passages.text: Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
negative_passages.docid: 3109
negative_passages.text: Prove that $(p+q)^m \leq p^m+q^m$
negative_passages.docid: 103496
negative_passages.text: Show that $3p^2=q^2$ implies $3|p$ and $3|q$
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$