Prove that $(a_1 − 1)(a_2 − 2)...(a_9 − 9)$ is always an even number.
42744
Prove a number is even using the Pigeonhole Principle
29545
$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
119282
If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.
106981
Prove that $(a+1)(a+2)...(a+b)$ is divisible by $b!$
51706
If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
94570
Proof that every odd integer is a difference of two squares
151520
About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
48430
Prove the following sequence always results in a perfect square.
45090
Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
6777
Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
130027
Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
55428
Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$
149716
Prove that $ \sqrt{n(n+1)(n+2)(n+3)+1} $ is always a natural number
100558
Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
83307
If $n\mid a^2-1$, $n$ an odd integer, show $n = \gcd(a-1, n)\gcd(a+1, n)$
91105
For every $n\ge3$ there exists $n$ different integers $a_1,...,a_n$ so that $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$