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shadow10
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14 votes

$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

11 votes

prove $ab\in A$, if $A=\{x^3+y^3+z^3-3xyz\mid x,y,z\in \mathbb Z\}$, $a,b\in A$

9 votes
Accepted

prove $\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$

8 votes
Accepted

How prove $x^3+y^3+z^3-3xyz\ge C|(x-y)(y-z)(z-x)|$

4 votes
Accepted

If $abc\neq 0$, then $ \frac{(a+b)^2}{c^2}+\frac{(a+c)^2}{b^2}+\frac{(b+c)^2}{a^2}\geq2 $

3 votes

Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$

2 votes
Accepted

How can I show the complete symmetric quadratic form has no zeros?

2 votes

What is the smallest integer $n$ greater than $1$ such that the root mean square of the first $n$ integers is an integer?

2 votes
Accepted

Proving that $(abc)^2\geq\left(\frac{4\Delta}{\sqrt{3}}\right)^3$, where $a$, $b$, $c$ are the sides, and $\Delta$ the area, of a triangle

2 votes

Can this limit be solved in this manner?

2 votes
Accepted

Find all the real roots of $P(Q(x))=0$

1 vote

is $A=\{v|v_i\leq v_{i+1},\forall i\in\{1,2...,n-1\}\}\subset\mathbb{R}^n$ clopen in $\mathbb{R}^n$

1 vote

Show $\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}$ for $a,b,c>0$

1 vote

Congruence of nth degree

1 vote

An inequality of riemann zeta function

1 vote

Prove $(a,b,c)=((a,b),(a,c))$

1 vote

You use Rolle's theorem to prove n-polynomial has n distinct roots, but how would you do this question?

1 vote

How can I find integers $n \gt 1$ such that the average of $1^2,2^2,3^2...n^2$ is itself a perfect square.

1 vote

A theory of equation question from my exam paper

1 vote

Prove the Sequence of Inequalities

1 vote

How to derive these inequalities?

1 vote

A problem of sum floors

1 vote
Accepted

Integer solution of $x^5+31=y^2$

1 vote

How to find a pattern in this recursive sequence algorithmically?

1 vote

If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

0 votes

Find the number of distinct elements.

0 votes

When is the product of two points on a circle is still on the circle?

0 votes

Trigonometry - How do I simplify this expression?

0 votes

What is $(p-1)!$ mod $(1 + 2 + \cdots + (p-1))$ where $p$ is an odd prime? (Exam Q)

0 votes

Define S as a set of primes such that if a, b are in S, ab+4 is in S. Show that S must be empty.