Articles of algebra precalculus

Proving that one of $a(1-b), b(1-c), c(1-a) \le \frac{1}{4}$

how can a prove that at least one of those is less than or equal to 1/4. $$\forall a,b,c\in \mathbb R^+, \ a(1-b)\leq 1/4 \lor b(1-c) \leq 1/4 \lor c(1-a) \leq 1/4.$$ help please!

Largest Triangular Number less than a Given Natural Number

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be written as $$57=T_{10}+2$$ The number $54$ can be written as $$54=T_{9}+9\neq T_{10}-1$$ The second part highlights that I am looking for Triangular numbers larger than a […]

Factoring with fractional exponents

I really hate to keep asking questions but I just can’t figure this out, I don’t know what is wrong with me but I can’t figure it out. I stared at it for 5 minutes and not a thought came into my head on how to do it that actually accomplished anything. $3x^{3/2}-9x^{1/2}+6x^{-1/2}$ I am […]

Easy way to find roots of the form $qi$ of a polynomial

Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem). Is there an easy way to check if $p$ have any roots of the form $qi$ where $q\in\mathbb{Q}$ (or at least $q\in\mathbb{Z}$) ? ($i\in\mathbb{C}$) ?

Zeilbergers algorithm in Maple

I try to prove several hard combinatorial identities. One of them is following \begin{align*} \sum_{s=0}^{\min\{k,n-1\}} \sum_{i=0}^{k-s} (-1)^{i} {2n+k-i-1 \choose k-s-i} {i-n \choose s} {n+i-1 \choose i} {n+k-s-1 \choose k} =\\ =\sum_{j=0}^{[\frac k2]} \sum_{i=0}^{min\{j, n-1\}}{ n-1 \choose i}^2 {{2n+j-i-2} \choose j-i} { n+k-2j-1 \choose n-1} .\text{ ($n,k$ are nonnegative integer)} \end{align*} Using the identity of Le-Jen […]

Factorise $y^2 -3yz -10z^2$

How do I solve this question? I have looked at the problem several times. However, I cannot find a viable solution. I believe that it is a perfect square trinomial problem.

Polynomial division problem

Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $ Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$. I’m not sure how to do this, as the only way I know of dividing polynomials other than long division is synthetic division, which only works with linear divisors. I thought about […]

Sum of first n natural numbers proof

I know how to prove this by induction but the text I’m following shows another way to prove it and I guess this way is used again in the future. I’m confused by it. So the expression for first n numbers is: $$\frac{n(n+1)}{2}$$ And this second proof starts out like this. It says since: $$(n+1)^2-n^2=2n+1$$ […]

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I’m really stuck on this one. Tried some stuff involving QM> AM(because the are positive) but can’t derive the needed ,can’t proceed from it.

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$. I am trying by calculus but don’t know how to use here in this problem. Any idea?