We have your learning goals covered with Spring and Summer courses available. Enroll today!

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
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Interesting NT function
amogususususus   5
N 11 minutes ago by AshAuktober
Source: Indonesia TSTST - Number Theory
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.

Proposed by IndoMathXdZ
5 replies
amogususususus
Feb 29, 2024
AshAuktober
11 minutes ago
Find min
hunghd8   1
N 13 minutes ago by Mathzeus1024
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
1 reply
hunghd8
2 hours ago
Mathzeus1024
13 minutes ago
4 free variables ineq
RainbowNeos   0
15 minutes ago
Given $0\leq a,b,c,d\leq 1$, show that
\[abc\sqrt{1-d}+bcd\sqrt{1-a}+cda\sqrt{1-b}+dab\sqrt{1-c}\leq 1.\]
0 replies
RainbowNeos
15 minutes ago
0 replies
An interesting inequality
JK1603JK   0
29 minutes ago
Source: unknown
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
0 replies
JK1603JK
29 minutes ago
0 replies
Romania Junior TST 2021 Day 3 P2
oVlad   4
N an hour ago by DensSv
Let $O$ be the circumcenter of triangle $ABC$ and let $AD$ be the height from $A$ ($D\in BC$). Let $M,N,P$ and $Q$ be the midpoints of $AB,AC,BD$ and $CD$ respectively. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be the circumcircles of triangles $AMN$ and $POQ$. Prove that $\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset$.
4 replies
oVlad
Jun 7, 2021
DensSv
an hour ago
A lot of z
Anulick   4
N an hour ago by quasar_lord
Source: CMI 2024
(a) FInd the number of complex roots of $Z^6 = Z + \bar{Z}$
(b) Find the number of complex solutions of $Z^n = Z + \bar{Z}$ for $n \in \mathbb{Z}^+$
4 replies
Anulick
May 19, 2024
quasar_lord
an hour ago
Inequalities
sqing   29
N an hour ago by SomeonecoolLovesMaths
Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=1. $ Prove that
$$(a^2-a+1)(b^2-b+1) \geq 9$$$$ (a^2-a+b+1)(b^2-b+a+1) \geq 25$$Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=\frac{2}{3}. $ Prove that
$$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$$$$(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$$
29 replies
sqing
Mar 10, 2025
SomeonecoolLovesMaths
an hour ago
2019 Chile Classification / Qualifying NMO Juniors XXXI
parmenides51   6
N an hour ago by bhontu
p1. Consider the sequence of positive integers $2, 3, 5, 6, 7, 8, 10, 11 ...$. which are not perfect squares. Calculate the $2019$-th term of the sequence.


p2. In a triangle $ABC$, let $D$ be the midpoint of side $BC$ and $E$ be the midpoint of segment $AD$. Lines $AC$ and $BE$ intersect at $F$. Show that $3AF = AC$.


p3. Find all positive integers $n$ such that $n! + 2019$ is a square perfect.


p4. In a party, there is a certain group of people, none of whom has more than $3$ friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
6 replies
parmenides51
Oct 11, 2021
bhontu
an hour ago
Inspired by Titu Andreescu
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c\geq 3abc . $ Prove that
$$a^2+b^2+c^2+1\geq \frac{4}{3}(ab+bc+ca) $$
2 replies
sqing
5 hours ago
sqing
an hour ago
INMO Problem 1
JetFire008   0
an hour ago
Source: INMO 2007
Problem 1: In a triangle $ABC$ right-angled at $C$, the median through $B$ bisects the angle between $BA$ and the bisector of $\angle B$. Prove that $\frac{5}{2}<\frac{AB}{BC}<3$.
0 replies
JetFire008
an hour ago
0 replies
Inequalities
sqing   12
N an hour ago by sqing
Let $ a,b $ be real numbers such that $ a + b  \geq  |ab + 1|. $ Prove that$$ a^3 + b^3 \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 2(a + b ) \geq  |ab + 1|. $ Prove that$$26( a^3 + b^3) \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 4(a + b) \geq 3|ab + 1|. $ Prove that$$148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$$
12 replies
sqing
Mar 8, 2025
sqing
an hour ago
Show these 2 circles are tangent to each other.
MTA_2024   0
an hour ago
A, B, C, and O are four points in the plane such that
\(\angle ABC > 90^\circ\)
and
\( OA = OB = OC \).

Let \( D \) be a point on \( (AB) \), and let \( (d) \) be a line passing through \( D \) such that
\( (AC) \perp (DC) \)
and
\( (d) \perp (AO) \).

The line \( (d) \) intersects \( (AC) \) at \( E \) and the circumcircle of triangle \( ABC \) at \( F \) (\( F \neq A \)).

Show that the circumcircles of triangles \( BEF \) and \( CFD \) are tangent at \( F \).
0 replies
MTA_2024
an hour ago
0 replies
Coordinate Geometry
JetFire008   0
an hour ago
Find the equations of the diagonals formed by the lines $2x-y+7=0, 2x-y-5=0, 3x+2y-5=0$ and $3x+2y+4=0$.
0 replies
JetFire008
an hour ago
0 replies
CMI just asks Schur directly
Anulick   6
N an hour ago by quasar_lord
(a) For non negetive $a,b,c, r$ prove that
\[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \](b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side.
(c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$
6 replies
Anulick
May 19, 2024
quasar_lord
an hour ago
Inequalites
Mario16   17
N Mar 17, 2025 by sqing
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
17 replies
Mario16
Feb 1, 2021
sqing
Mar 17, 2025
Inequalites
G H J
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.
Mario16
100 posts
#1 • 1 Y
Y by Mango247
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
IAmTheHazard
5000 posts
#2
Y by
You literally posted this 2 hours ago.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Mario16
100 posts
#3
Y by
Yes but i forgot to write Something
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
IAmTheHazard
5000 posts
#4
Y by
It seems to be the exact same to me.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Wildabandon
506 posts
#5
Y by
Mario16 wrote:
Yes but i forgot to write Something

You can edit your post.

If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
This post has been edited 1 time. Last edited by Wildabandon, Feb 2, 2021, 12:23 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Quantum_fluctuations
1282 posts
#6 • 1 Y
Y by Mango247
This is where you should go.
https://artofproblemsolving.com/community/c6h2387664
This post has been edited 1 time. Last edited by Quantum_fluctuations, Feb 2, 2021, 12:23 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
KP9
27 posts
#7
Y by
Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

So we have : $\sum \frac{1}{5+a^2} \leq \frac{3}{5} - \frac{1}{90}(a^3+b^3+c^3) - \frac{1}{45}(a+b+c) = \frac{3}{5} -\frac{3}{45} - \frac{1}{90}(a^3+b^3+c^3) = \frac{8}{15} - \frac{a^3+b^3+c^3}{90}$ (1)

We also have : $(1+1+1)(1+1+1)(a^3+b^3+c^3)\geq (a+b+c)^3$

$\Rightarrow a^3 + b^3 + c^3 \geq 3$ (2)

(1) and (2) $\Rightarrow \sum \frac{1}{5+a^2} \leq \frac{8}{15} - \frac{3}{90} = \frac{1}{2}$ ( Q.E.D)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Quantum_fluctuations
1282 posts
#8
Y by
KP9 wrote:
Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

How did you find that?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
KP9
27 posts
#10 • 1 Y
Y by Mango247
oh , iam sorry , i have a problem when i prove it
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
logrange
120 posts
#11 • 1 Y
Y by Mango247
I have another similar problem in which sum is cyclic in one variable only. In these kind of problems we can use tangent line, but I want to know that whether it can be used in problems which have expression ≤ constant. I have used it only in cases like expression ≥ constant. If you can do this by tangent line then please post the solution of this by tangent line also.
Prove that cyclic sum $\frac{a}{2a^2+a+1}\leq \frac{3}{4}$
Given a+b+c=3 (Sorry, I missed that earlier)
This post has been edited 3 times. Last edited by logrange, Feb 2, 2021, 5:39 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
logrange
120 posts
#12
Y by
Bump bump bump
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
logrange
120 posts
#13
Y by
Anyone?
Note - I want a solution without n-1 EV (Calculus)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Wildabandon
506 posts
#14
Y by
I'm thinking Jensen but still using the calculus LOL
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
logrange
120 posts
#15
Y by
@below
Thanks
This post has been edited 1 time. Last edited by logrange, Feb 3, 2021, 11:13 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
starchan
1601 posts
#16
Y by
I think Chebyshev's kills this one..
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41150 posts
#17
Y by
Wildabandon wrote:
If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
https://artofproblemsolving.com/community/c4h2391785p19635446
https://artofproblemsolving.com/community/c6h2387664p20486022
Let $a,b,c$ be non-negative numbers such that $ab+bc+ca+abc=4.$ Prove that
$$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}= 1$$$$\frac{1}{2}\leq \frac{1}{a^2+4}+\frac{1}{b^2+4}+\frac{1}{c^2+4}\leq \frac{3}{5}$$https://artofproblemsolving.com/community/c6h1510436p8962718
Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$\frac{1}{a^2+5}+\frac{1}{b^2+5}+\frac{1}{c^2+5}\leq \frac{1}{2}$$( Vasile Cîrtoaje)
$$a^2b+b^2c+c^2a\leq \frac{368}{3}-\frac{176\sqrt{33}}{9}$$https://artofproblemsolving.com/community/c6h382474p2119615
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 2:55 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41150 posts
#18
Y by
Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$ ab+bc+ca-  abc\leq 2$$$$\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\geq \frac{3}{2}$$$$\frac{1}{a^2+8}+\frac{1}{b^2+8}+\frac{1}{c^2+8}\leq \frac{1}{3}$$
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 3:30 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41150 posts
#19
Y by
Let $ a,b\geq 0 $ and $a+b+a^2+ab+b^2 =5.$ Prove that
$$ \frac{1}{a^2+1}+ \frac{1}{b^2+1}  \geq1$$$$  \frac{1}{a^2+2}+ \frac{1}{b^2+2} \geq \frac{2}{3}$$$$  \frac{1}{a^2+\frac{53}{20}}+ \frac{1}{b^2+\frac{53}{20}} \geq \frac{40}{73}$$$$  \frac{1}{a^2+\frac{1327}{500}}+ \frac{1}{b^2+ \frac{1327}{500}} \geq \frac{1000}{1827}$$$$ \frac{1}{a^2+3}+ \frac{1}{b^2+3}  \geq \frac{185+3\sqrt{21}}{402}$$
Z K Y
N Quick Reply
G
H
=
a