query: Proving $(1+a_1)(1+a_2)...(1+a_n) \geq 2^n$
positive_passages.docid: 21986
positive_passages.text: What is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$?
negative_passages.docid: 67784
negative_passages.text: If $A + A^t = 2I$ then $\det(A) \geq 1$
negative_passages.docid: 88687
negative_passages.text: Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
negative_passages.docid: 73987
negative_passages.text: Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
negative_passages.docid: 105641
negative_passages.text: Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 83662
negative_passages.text: Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
negative_passages.docid: 139503
negative_passages.text: How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
negative_passages.docid: 69921
negative_passages.text: Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
negative_passages.docid: 111316
negative_passages.text: PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
negative_passages.docid: 27570
negative_passages.text: Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
negative_passages.docid: 110790
negative_passages.text: Prove the following summation inequality: $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$
negative_passages.docid: 99692
negative_passages.text: Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
negative_passages.docid: 67527
negative_passages.text: Proof by Induction that $3^n ≥ 1+2^n$
negative_passages.docid: 32149
negative_passages.text: Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
negative_passages.docid: 64096
negative_passages.text: Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$