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 40 Coffee Break Riddle 13 Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ 11 Let $a,b,c >0$ Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$ 11 How do I solve this analytically $3^x=9x$ 10 prove for all $n\geq 0$ that $3 \mid n^3+6n^2+11n+6$

### Reputation (8,109)

 +5 Is $f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$? +10 If $m,n \in \mathbb{N}$ and $m> n$ then $(a^{2^{n}}+1)|(a^{2^{m}}-1)$ +5 $3 \times 3$ Magic Square of Squares +10 Find all positive integers $n$ such that $\phi(n)=6$.

### Questions (100)

 46 Is $f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$? 20 Infinitely many primes of the form $\lfloor \sqrt {3} \cdot n \rfloor$? 16 Are polynomials of the form : $f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$ irreducible over $\mathbb{Z}$? 13 Approximation of $e$ using $\pi$ and $\phi$? 11 A theorem about prime divisors of generalized Fermat numbers?

### Tags (123)

 75 algebra-precalculus × 47 41 puzzle × 8 60 geometry × 50 33 limits × 16 57 elementary-number-theory × 86 31 differential-equations × 22 56 calculus × 51 26 trigonometry × 14 45 recreational-mathematics × 9 26 discrete-mathematics × 9

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