Inequalities

Art of Problem Solving

AoPS Online

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

G

Topic

First Poster

Last Poster

k a March Highlights and 2025 AoPS Online Class Information

jlacosta 0

Mar 2, 2025

March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)

0 replies

jlacosta

Mar 2, 2025

0 replies

Complex + Radical Evaluation

Saucepan_man02 1

N a minute ago by MathRook7817

Evaluate: (without calculators)
$(\sqrt{6 - 2 \sqrt{5}} + i \sqrt{2 \sqrt{5} + 10})^5 + (\sqrt{6 - 2 \sqrt{5}} - i \sqrt{2 \sqrt{5} + 10})^5$

1 reply

Saucepan_man02

23 minutes ago

MathRook7817

a minute ago

Inequality with integers and indices

Michael Niland 1

N 22 minutes ago by invisibleman

Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$ , and for a positive integer $n\geq3$ , prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$ .

1 reply

Michael Niland

Today at 6:52 AM

invisibleman

22 minutes ago

Inequalities

sqing 6

N 4 hours ago by sqing

Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 3$ . Prove that
$a+b+a^2+b^2 \geq 28-6\sqrt{21}$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 4$ . Prove that
$a+b+a^2+b^2 \geq 10-4\sqrt 5$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 5$ . Prove that
$a+b+a^2+b^2 \geq \frac {2(26-5\sqrt{13})}{9}$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1}\geq 3+ \sqrt {5}$ . Prove that
$a+b+a^2+b^2 \geq 2$

6 replies

sqing

Mar 14, 2025

sqing

4 hours ago

Inequalities from SXTX

sqing 10

N Today at 7:30 AM by sqing

T702. Let $a,b,c>0$ and $a+2b+3c=\sqrt{13}.$ Prove that $\sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$ S
T703. Let $a,b$ be real numbers such that $a+b\neq 0.$ . Find the minimum of $a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $a,b,c>0$ and $a+b+c=3.$ Prove that $\frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$ S

10 replies

sqing

Feb 18, 2025

sqing

Today at 7:30 AM

Inequality with roots of cubic

Michael Niland 0

Today at 7:23 AM

The equation $x^3+px+q =0$ has real roots $a_1 ,a_2 , a_3$ where $a_1 \leq a_2 \leq a_3$ .

Similarly the equation $x^3 +rx +s=0$ has real roots $b_1, b_2, b_3$ where $b_1 \leq b_2,\leq b_3$ .

Prove that if $\frac{a_1}{b_1} \leq \frac{a_2}{b_2} \leq \frac{a_3}{b_3}$ (s non zero), then $(\frac{p}{r})^3 =(\frac{q}{s})^2$

0 replies

Michael Niland

Today at 7:23 AM

0 replies

Inequalities

nhathhuyyp5c 3

N Today at 6:36 AM by giangtruong13

Let $a,b,c$ be positive reals such that $a+b+c+2=abc$ . Find the maximum value of $\frac{a+1}{a^2+2}+\frac{b+1}{b^2+2}+\frac{c+1}{c^2+2}$ Given $n\ge 2$ non-zero reals $x_1,x_2,\cdots x_n$ such that their sum is $100$ . Prove that there exists two numbers $x_i,x_j$ such that $\frac{1}{2}\le \left|\frac{x_i}{x_j}\right|\le 2$

3 replies

nhathhuyyp5c

Jan 10, 2025

giangtruong13

Today at 6:36 AM

Algebra-1

JetFire008 3

N Today at 5:11 AM by BackToSchool

Find real numbers $p$ , $q$ if $1+i$ is a root of $x^3+px^2+qx+6=0$ and solve the equation.

3 replies

JetFire008

Yesterday at 3:19 PM

BackToSchool

Today at 5:11 AM

Cool problem

jkim0656 3

N Today at 4:09 AM by RedFireTruck

Hey AoPS!
I came across a problem recently and it goes like this:
Which is greater:
$2 ^ {100!}$ or $2^{100}!$ Soo... can u guys help? thx!
:yoda: and may the force be with u!
Notes: I don't exactly know where this problem came from, but if u find that u are the orig maker of this problem feel free to drop me a PM and ill add u to my post :)

3 replies

jkim0656

Today at 3:04 AM

RedFireTruck

Today at 4:09 AM

help me at this pls i'm so confused

myname17042005 12

N Today at 3:07 AM by sqing

if a, b, c are real numbers and a, b, c >0, prove that
pls help meee

12 replies

myname17042005

Jun 12, 2020

sqing

Today at 3:07 AM

Trig Multiplication

szhang7 1

N Today at 2:24 AM by joeym2011

Find the exact value of $(2-\sin^2(\frac{\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))$ .

1 reply

szhang7

Yesterday at 3:50 PM

joeym2011

Today at 2:24 AM

Algebra-2

JetFire008 1

N Today at 2:11 AM by joeym2011

Prove that if $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$ , then $f(x^n)$ is divisible by $x^n-1$ .

1 reply

JetFire008

Yesterday at 3:23 PM

joeym2011

Today at 2:11 AM

a/b + b/a never integer ?

MTA_2024 3

N Yesterday at 10:35 PM by ohiorizzler1434

Let $a$ and $b$ be 2 distinct positive integers.
Can $\frac a b +\frac b a$ be in an integer. Prove why ?

3 replies

MTA_2024

Yesterday at 3:08 PM

ohiorizzler1434

Yesterday at 10:35 PM

2014 Community AIME / Marathon ... Algebra Medium #1 quartic

parmenides51 5

N Yesterday at 7:17 PM by CubeAlgo15

Let there be a quartic function $f(x)$ with maximums $(4,5)$ and $(5,5)$ . If $f(0) = -195$ , and $f(10)$ can be expressed as $-n$ where $n$ is a positive integer, find $n$ .

proposed by joshualee2000

5 replies

parmenides51

Jan 21, 2024

CubeAlgo15

Yesterday at 7:17 PM

interesting problem

sausagebun 1

N Yesterday at 4:05 PM by mathprodigy2011

Six points, labeled A, B, C, D, E, and F, are positioned consecutively on a straight line. Let G be a point not located on this line. The following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. Determine the area of triangle BGE.
I brute forced this with trig, was wondering if theres a more elegant way of doing this

1 reply

sausagebun

Yesterday at 3:21 PM

mathprodigy2011

Yesterday at 4:05 PM

Inequalities

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

51 posts

#1 h Jan 10, 2025, 1:50 AM

Y by

Let $a,b,c$ be positive reals such that $a+b+c+2=abc$ . Find the maximum value of $\frac{a+1}{a^2+2}+\frac{b+1}{b^2+2}+\frac{c+1}{c^2+2}$ Given $n\ge 2$ non-zero reals $x_1,x_2,\cdots x_n$ such that their sum is $100$ . Prove that there exists two numbers $x_i,x_j$ such that $\frac{1}{2}\le \left|\frac{x_i}{x_j}\right|\le 2$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

41042 posts

sqing

#2 h Jan 10, 2025, 1:58 AM

Y by

Let $a,b,c$ be positive reals such that $a+b+c+2=abc$ . Prove that $\frac{a+1}{a^2+2}+\frac{b+1}{b^2+2}+\frac{c+1}{c^2+2}\leq \frac32$ *

Attachments:

This post has been edited 1 time. Last edited by sqing, Today at 4:42 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

51 posts

#3 h Today at 2:52 AM

Y by

Let $a,b,c$ be positive reals such that $a+b+c+2=abc$ . Prove that $a^2+b^2+c^2\geq 2(\sum{\sqrt{ab}})$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

59 posts

#4 h Today at 6:36 AM

Y by

There will exist $a=\frac{y+z}{x},b=\frac{z+x}{y},c=\frac{x+y}{z} (x,y,z>0)$ by using the condition $a+b+c+2=abc$

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a