5
votes
4answers
96 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
3
votes
2answers
59 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
1
vote
2answers
77 views

Let $a,b,c \in \mathbb{R^+}$, does this inequality holds $\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$?

Does the following statement/inequality holds for $a,b,c \in \mathbb{R^+}$? $$\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$$ I've been thinking for hours and I ...
3
votes
1answer
133 views

A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$.

Let $P(x)$ be a polynomial with only real roots and all coefficients equal to $\pm 1$. Prove that the degree of the polynomial is less than 4. This is practice for Putnam, but I am not certain where ...
1
vote
2answers
116 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
9
votes
4answers
986 views

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + ...
15
votes
4answers
594 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
1
vote
1answer
37 views

Maximal Distinct Roots in $F_q$

Let $a\in F_q[x]$, and let $r(\cdot)$ denote the number of distinct roots over $F_q$. For any $i|q$, prove that $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$
1
vote
1answer
52 views

Try to solve root in inequality got wrong result

I am very confused. So I have to solve this inequality. The result is $13/24$. But if I try to solve it myself, I get $17/24$. Because: $$\sqrt{\left(\frac{-5}{24}\right)^2 + \frac{1}{4}} = ...
2
votes
1answer
181 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
38
votes
4answers
941 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...