Finding $n$ such that in a regular $n$-gon $A_1A_2\ldots A_n$ we have $\frac1{A_1A_2}=\frac1{A_1A_3}+\frac1{A_1A_4}$
123475
$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$.Then find the value of $n$
70620
From $n=2$ to $n$-ary Euclid lemma $\ p\mid a_1\cdots a_n\Rightarrow\ p\mid a_1\,$ or $\,\cdots\, p\mid a_n$
16382
Find $a$ such that $(x+a)(x+1991)+1=(x+b)(x+c)$ with $a,b,c\in\Bbb Z$
8666
Let us define a series $\{a(n)\}$ such that $a(n)= a(n-1) + \frac{1}{a(n-1)}$
108310
$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
149700
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
41445
Show that the sequence $a_n\leq a_{2n}+a_{2n+1}$ diverges
68771
Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.
156032
Find a such that $112_a$, $63_a$ and $34_a$ are in an arithmetic progression
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
25648
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
83307
If $n\mid a^2-1$, $n$ an odd integer, show $n = \gcd(a-1, n)\gcd(a+1, n)$
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.
57503
How find $\sum_{k \in A} \frac{1}{k-1} $ for $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $
97805
$n+1$ is a divisor of $\binom{2n} {n}$
31196
Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
55428
Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$