Inequalities

by sqing, May 3, 2025, 3:57 AM

Let $ a,b,c\geq 0 ,a+b+c =4. $ Prove that
$$2a +ab +ab^2c \leq\frac{63+5\sqrt 5}{8}$$$$2a +ab^2 +abc \leq \frac{4(68+5\sqrt {10})}{27}$$$$ 2a +a^2b + a b^2c^3\leq \frac{4(50+11\sqrt {22})}{27}$$
inequalities
L

Inequalities

by toanrathay, May 3, 2025, 3:22 AM

Let \( a, b, c > 0 \) such that $2(b^2 + bc + c^2) = 1 - 3a^2,$ prove that
\[ a + b + c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 10. \]
inequalities
L

Inequalities

by sqing, May 3, 2025, 3:02 AM

Let $ a,b \geq 0 $ and $ a-b+a^3-b^3=2 $.Prove that$$ a^2+ab+b^2 \geq 1 $$Let $ a,b \geq 0 $ and $ a+b+a^3+b^3=2 $.Prove that$$ a^2-ab+b^2 \leq 1 $$
This post has been edited 1 time. Last edited by sqing, Today at 3:05 AM
inequalities
L

Floor and exact value

by Ecrin_eren, May 2, 2025, 5:54 PM

The exact value of a real number a is denoted by [a] and
the fractional value {a}.
For example; [3.7]= 3 and {3, 7} = 0.7
For a positive real number x,
Given the equality of [x]{x} = 2023, what can
[X^2]-[x]^2 be?
This post has been edited 1 time. Last edited by Ecrin_eren, Yesterday at 5:56 PM
Reason: Mb
algebra
L

How many pairs

by Ecrin_eren, May 2, 2025, 3:08 PM

Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:

p + 2^p + 3 = n^2 ?
number theory
L

Find the minimum

by Ecrin_eren, May 1, 2025, 9:03 PM

The polynomial is given by P(x) = x^4 + ax^3 + bx^2 + cx + d, and its roots are x1, x2, x3, x4. Additionally, it is stated that d ≥ 5.Find the minimum value of the product:

(x1^2 + 1)(x2^2 + 1)(x3^2 + 1)(x4^2 + 1).
algebra
L

Find the functions

by Ecrin_eren, May 1, 2025, 8:58 PM

"Find all differentiable functions f that satisfy the condition f(x) + f(y) = f((x + y) / (1 - xy)) for all x, y ∈ R, where xy ≠ 1."
This post has been edited 1 time. Last edited by Ecrin_eren, Yesterday at 9:07 AM
Reason: Mb
algebra
function
L

Range of a trigonometric function

by Saucepan_man02, Apr 28, 2025, 4:44 PM

Find the range of the function: $f(x)=\frac{\sin^2 x+\sin x-1}{\sin^2 x-\sin x+2}$.
function
L

Inequalities

by sqing, Apr 26, 2025, 12:58 PM

Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
inequalities
algebra
L

Calculating combinatorial numbers

by lgx57, Mar 30, 2025, 8:20 AM

Try to simplify this expression:

$$\sum_{i=1}^n \sum_{j=1}^i C_{n}^i C_{n}^j$$
algebra
combinatorics
L

Fun with math!

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  • Please stop accusing aoum of AI. Even if someone has phenomenal conventions and style, that does not imply that they used AI. I don't want to discuss this, but I do want to stop krish6_9 from accusing aoum of AI. In general, please don't accuse aoum of AI

    by HacheB2031, 4 hours ago

  • bro this is 100% ai generated lil bro ur not slick

    by krish6_9, Today at 2:10 AM

  • If you leave a comment on one of my posts—especially older ones—I might not see it right away.

    by aoum, Yesterday at 11:55 PM

  • Thank you, HacheB2031. :) I want no further discussion of AI-generated content in the shout-box.

    by aoum, Yesterday at 12:43 AM

  • why would aoum use ai in his or her blog, he or she seems like a nice person :(

    by HacheB2031, May 1, 2025, 3:15 AM

  • Any unfounded allegations regarding AI-generated content violate Pi in the Sky blog standards. Continued infractions will result in disciplinary action, including bans, in accordance with platform guidelines. This is a formal warning.

    by aoum, Apr 27, 2025, 11:19 PM

  • It would be rude to call this AI-generated if it was not. But I find the title (in blog post), organization, and general word choices very suspicious

    by RubixMaster21, Apr 27, 2025, 1:25 AM

  • um this does seem slightly similar to ai

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  • 100 posts!

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  • Very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very cool (The maximum of the factorial machine is 7228！

    by Coin1, Apr 21, 2025, 4:44 AM

  • cool blog and good content but it looks eerily similar to chatgpt

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  • 1,000 views!

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  • Are you asking to contribute or to be notified whenever a post is published?

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