mathh

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78 Questions

 14 Prove that if $x_1,x_2,\ldots,x_n>0$, then $(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$. 10 If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. 10 Find all $x,y,z\in\mathbb N$, $x,y,z>1$ such that satisfy $x\mid yz+1$, $y\mid xz+1$, and $z\mid xy+1$ 10 If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$. 8 Prove that there are no such different positive numbers that satisfy both $a+b=c+d$ and $a^3+b^3=c^3+d^3$.

1,900 Reputation

 +2 Symbols for “odd” and “even” +10 Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)? +10 When an equation has no solutions, denote it with $x\in\varnothing$. +2 By induction, show that for ∀n∈N, it is true that:

 10 How to deduce a closed formula given an equivalent recursive one? 6 If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$. 4 prove that $\frac{ab}{a+b}+\frac{ac}{a+c}+\frac{ad}{a+d}+\frac{bc}{b+c}+\frac{bd}{b+d}+\frac{cd}{c+d}\le \frac{3}{4}$ 4 Concepts in mathematics which are referred to as 'generalizations' 4 what would be the way to solve a system of equations like this one?

56 Tags

 10 closed-form 4 inequality × 20 10 recurrence-relations 4 systems-of-equations × 2 8 elementary-number-theory × 35 4 quadratic-forms × 2 6 roots × 5 4 quadratics 6 calculus × 3 4 quadratic-equation

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