Inequality

by lgx57, Aug 3, 2025, 5:41 AM

$a,b,c \in(0,1),ab+bc+ca=1$ ,Prove that:

$\dfrac{8\sqrt{3}}{9}\le (a+b)(b+c)(c+a) < 2$
(Not use trigonometric commutator)

inequalities

A Test Pre IMO ViệtNam

by Math2030, Aug 3, 2025, 1:40 AM

Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n$ not is a perfect square for all $n > 3$ .

Circle geometry proof

by littleduckysteve, Aug 2, 2025, 7:29 AM

Suppose 3 circles are drawn in the 2-dimensional grid such that no two circles are of the same radius. Now we draw the 2 lines which are both tangent to the smallest circle and the median circle, and call their intersection, A. Now we do the same thing for the biggest circle and the smallest, and finally the biggest and the median circles. Now assume that we call these two points, B and C. Prove that A, B, and C are all colinear regardless of where the circles are.

Inequalities

by sqing, Aug 1, 2025, 12:19 PM

Let $a,b \geq 0,a +b =1 .$ Prove that
$\sqrt{a^4 + \frac{3}{4}ab} + \sqrt{b^4 + \frac{3}{4}ab} \geq 1$ Let $a,b \geq 0,a +b =4 .$ Prove that
$\sqrt{a^4 + 12ab} + \sqrt{b^4 + 12ab} \geq 16$ Let $a,b,c \geq 0,a +b +c=1 .$ Prove that
$a^2 + b^2 + c^2 + \frac{1}{2}\sqrt{3abc} \geq \frac{1}{2}$ Let $a,b,c \geq 0,a +b +c=3 .$ Prove that
$a^2 + b^2 + c^2 +\frac{3}{2} \sqrt{abc} \geq \frac{9}{2}$

This post has been edited 3 times. Last edited by sqing, Aug 1, 2025, 12:37 PM

inequalities

Inequalities

by sqing, Jul 31, 2025, 12:30 PM

Let $a,b \geq 0, 2a +b^2=1 .$ Prove that
$\frac{\sqrt{a+b}}{a(b+1)} \geq \sqrt{2}$ Let $a,b \geq 0, a^2 +2b^2 =2 .$ Prove that
$\frac{\sqrt{a+2b}}{a(b+1)} \geq \frac{9}{8\sqrt{2}}$ Let $a,b \geq 0, 2a^2 +b^2 =1 .$ Prove that
$\dfrac{a^2}{a+b}\leq\frac{1}{\sqrt{2}}$ $\frac{7}{10}>\frac{a+b}{a^2+b^2+1} \geq\frac{\sqrt{2}}{3}$

This post has been edited 3 times. Last edited by sqing, Jul 31, 2025, 1:18 PM

inequalities

one problem from Mathworks 2023 contest

by Carmen8102, Jul 30, 2025, 11:06 PM

The letters A and B represent different digits. The 5-digit integer AB5AB = X^3-3X^2 for some two-digit integer X. Find X.

Is there a faster way of solving this problem, not trying so many numbers?

This post has been edited 1 time. Last edited by Carmen8102, Yesterday at 4:08 PM
Reason: I changed the lowercase x to upper case X to make it consistent

Mathworks

Theorems that really helped

by SwordAxe, Jul 29, 2025, 4:01 PM

What are some theorems that really helped you in competitions (specifically AMC 8/10/12, AIME, Mathcounts)?

Herons really helped me once

Burning Bud on Stock

by lrnnz, Jul 21, 2025, 2:44 PM

Burning Bud is currently in stock in Grow a Garden, and you can either buy a pack of:
- 12 stocks of Burning Bud, or
- 5 stocks of Burning Bud.

Determine the maximum amount of Burning Bud you can't earn, after buying some stocks of it.
Answer: Click to reveal hidden text
Solution: Click to reveal hidden text

Chicken McNugget Theorem

grow a garden

Find the largest value of p

by Darealzolt, Jun 6, 2025, 4:24 PM

It is known that
$\sqrt{x-3}+\sqrt{6-x} \leq p$ In which $x \in \mathbb{R}$ , hence find the largest value of $p$ .

inequalities

15 dropped AIME problems from 1983-88 #1 37 | 37abc, 37bca, 37cab

by parmenides51, Jan 22, 2024, 2:23 PM

Determine the number of five digit integers $37abc$ (in base $10$ ), such that $37abc, 37bca, 37cab$ is divisible by $37$ .

number theory

Submit

another shout lol
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I'm still here guys!
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by player01, Jul 18, 2022, 10:41 PM
by SashaMath, Jun 12, 2022, 1:36 PM
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by priceyfalcon, Oct 10, 2021, 5:35 PM

17 shouts

Cookie Crumbs

by lgx57, Aug 3, 2025, 5:41 AM

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by sqing, Aug 1, 2025, 12:19 PM

by sqing, Jul 31, 2025, 12:30 PM

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by lrnnz, Jul 21, 2025, 2:44 PM

by Darealzolt, Jun 6, 2025, 4:24 PM

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