Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
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
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
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
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 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
IMO Shortlist 2011, G2
WakeUp   30
N 14 minutes ago by ezpotd
Source: IMO Shortlist 2011, G2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]

Proposed by Alexey Gladkich, Israel
30 replies
WakeUp
Jul 13, 2012
ezpotd
14 minutes ago
2-var inequality
sqing   1
N 33 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 ,\frac{a}{b+2}+\frac{b}{a+2}+ \frac{ab}{3}\leq 1.$ Prove that
$$ a^2+b^2 +\frac{5}{3}ab \leq 4$$
1 reply
sqing
an hour ago
sqing
33 minutes ago
Rational Points in n-Dimensional Space
steven_zhang123   0
an hour ago
Let \( T = (x_1, x_2, \ldots, x_n) \), where \( x_i \) is rational for \( i = 1, 2, \ldots, n \). A vector \( T \) is called a rational point in \( n \)-dimensional space. Denote the set of all such vectors \( T \) as \( S \). For \( A = (x_1, x_2, \ldots, x_n) \) and \( B = (y_1, y_2, \ldots, y_n) \) in \( S \), define the distance between points \( A \) and \( B \) as \( d(A, B) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2} \). We say that point \( A \) can move to point \( B \) if and only if there is a unit distance between two points in \( S \).

Prove:
(1) If \( n \leq 4 \), there exists a point that cannot be reached from the origin via a finite number of moves.
(2) If \( n \geq 5 \), any point in \( S \) can be reached from any other point via moves.
0 replies
steven_zhang123
an hour ago
0 replies
Inspired by old results
sqing   5
N 2 hours ago by sqing
Source: Own
Let $a,b,c $ be reals such that $a^2+b^2+c^2=3$ .Prove that
$$(1-a)(k-b)(1-c)+abc\ge -k$$Where $ k\geq 1.$
$$(1-a)(1-b)(1-c)+abc\ge -1$$$$(1-a)(1-b)(1-c)-abc\ge -\frac{1}{2}-\sqrt 2$$
5 replies
sqing
Yesterday at 7:36 AM
sqing
2 hours ago
equation in integers
Pirkuliyev Rovsen   2
N 2 hours ago by ytChen
Solve in $Z$ the equation $a^2+b=b^{2022}$
2 replies
Pirkuliyev Rovsen
Feb 10, 2025
ytChen
2 hours ago
Inequality for Sequences
steven_zhang123   0
3 hours ago
Given a positive number \( t \) and integers \( m, n \geq 2 \), prove that for any \( n \) positive numbers \( a_1, a_2, \ldots, a_n \) satisfying \( a_j - a_{j-1} \leq t \) for \( j = 1, 2, \ldots, n \) (with the convention \( a_0 = 0 \)), the following inequality holds:
\[
\sum_{j=1}^n a_j^{2m-1} \leq \frac{mt}{2} \left( \sum_{j=1}^n a_j^{m-1} \right)^2.
\]
0 replies
steven_zhang123
3 hours ago
0 replies
Circumcircle of ADM
v_Enhance   70
N 3 hours ago by Shan3t
Source: USA TSTST 2012, Problem 7
Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
70 replies
v_Enhance
Jul 19, 2012
Shan3t
3 hours ago
Prove this relation in triangle ABC
Entrepreneur   1
N 4 hours ago by MathIQ.
Source: Conjectured by me
In $\Delta ABC,$ prove that$$\color{blue}{2\angle A=3\angle B\implies c^3a^3(c+a)^2=b^2(c^2+a^2-b^2+2ca)(c^2+a^2-b^2)^2.}$$
1 reply
Entrepreneur
Aug 1, 2024
MathIQ.
4 hours ago
45 degrees everywhere
Rijul saini   12
N 4 hours ago by guptaamitu1
Source: India IMOTC 2024 Day 2 Problem 2
Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$.

Proposed by Atul Shatavart Nadig and Shantanu Nene
12 replies
Rijul saini
May 31, 2024
guptaamitu1
4 hours ago
equation 2025
mohamed-adam   1
N 4 hours ago by MathIQ.
Source: own
Find all positive integers $a,b$ such that $$a^8-2b^5=2025b$$
1 reply
mohamed-adam
Yesterday at 7:49 PM
MathIQ.
4 hours ago
Graph Theory
ABCD1728   1
N 4 hours ago by ABCD1728
Can anyone provide the PDF version of "Graphs: an introduction" by Radu Bumbacea (XYZ press), thanks!
1 reply
ABCD1728
Yesterday at 5:10 AM
ABCD1728
4 hours ago
Rectangles of grid cells
tapir1729   11
N 4 hours ago by Mathandski
Source: TSTST 2024, problem 9
Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$.

Anonymous
11 replies
tapir1729
Jun 24, 2024
Mathandski
4 hours ago
Interesting problem from a friend
v4913   11
N 5 hours ago by YaoAOPS
Source: I'm not sure...
Let the incircle $(I)$ of $\triangle{ABC}$ touch $BC$ at $D$, $ID \cap (I) = K$, let $\ell$ denote the line tangent to $(I)$ through $K$. Define $E, F \in \ell$ such that $\angle{EIF} = 90^{\circ}, EI, FI \cap (AEF) = E', F'$. Prove that the circumcenter $O$ of $\triangle{ABC}$ lies on $E'F'$.
11 replies
v4913
Nov 25, 2023
YaoAOPS
5 hours ago
Curious inequality
produit   2
N 5 hours ago by RainbowNeos
Positive real numbers x_1, x_2, . . . x_n satisfy x_1 + x_2 + . . . + x_n = 1.
Prove that
1/(1 −√x_1)+1/(1 −√x_2)+ . . . +1/(1 −√x_n)⩾ n + 4.
2 replies
produit
May 10, 2025
RainbowNeos
5 hours ago
Inspired by learningimprove
sqing   7
N Apr 19, 2025 by sqing
Source: Own
Let $ a,b,c,d\geq0, (a+b)(c+d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq1 $$Let $ a,b,c,d\geq0, (a+2b)(c+2d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq\frac{2}{5} $$Let $ a,b,c,d\geq0, (a+2b)(2c+ d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq\frac{3}{7} $$
7 replies
sqing
Apr 19, 2025
sqing
Apr 19, 2025
Inspired by learningimprove
G H J
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42357 posts
#1
Y by
Let $ a,b,c,d\geq0, (a+b)(c+d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq1 $$Let $ a,b,c,d\geq0, (a+2b)(c+2d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq\frac{2}{5} $$Let $ a,b,c,d\geq0, (a+2b)(2c+ d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq\frac{3}{7} $$
This post has been edited 1 time. Last edited by sqing, Apr 19, 2025, 7:54 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GeoMorocco
44 posts
#2
Y by
sqing wrote:
Let $ a,b,c,d\geq0, (a+b)(c+d)=2 . $ Prove that
$$  a^2+b^2+c^2+d^2-ac-bd \geq1 $$
we have:
$$ 2(a^2+b^2+c^2+d^2-ac-bd-1) =(a-c)^2 + (b-d)^2+(a^2+b^2+c^2+d^2-ac-ad-bc-bd)$$which is true by rearrangement. Equality for $a=b=c=d$.

The other inequalities can be proved in a similar way.
This post has been edited 1 time. Last edited by GeoMorocco, Apr 19, 2025, 8:08 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42357 posts
#3
Y by
Nice.Thanks.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42357 posts
#4
Y by
Let $ a,b,c $ be reals such that $ a+2b+c=0 $ and $ a^2+b^2+c^2=3. $ Prove that
$$abc\leq 1$$Let $ a,b,c $ be reals such that $ a+2b+c=0 $ and $ a^2+2b^2+c^2=3. $ Prove that
$$abc\leq\frac{3\sqrt{3}}{8}$$Let $ a,b,c $ be reals such that $ a+4b+c=0 $ and $ a^2+2b^2+c^2=3. $ Prove that
$$abc\leq\frac{3 }{5}\sqrt{\frac{6}{5}}$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SunnyEvan
129 posts
#5
Y by
$$ a+2b+c=0 \iff b= -\frac{a+c}{2}$$$$ a^2+b^2+c^2=3 \iff 5b^2-2ac=3 $$$$ abc \leq 1 \iff 5b^3-3b-2 \leq 0 \iff b \leq 1 \iff |a+c| \leq 2 $$because : $$ \frac{5}{4}(a+c)^2 \leq 3+ \frac{1}{2}(a+c)^2 \iff |a+c| \leq 2 $$The other inequalities can be proved in a similar way. :-D
equality case : $ (a,b,c)=(-1,1,-1)$
This post has been edited 2 times. Last edited by SunnyEvan, Apr 19, 2025, 10:45 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GeoMorocco
44 posts
#6
Y by
SunnyEvan wrote:
$$ a+2b+c=0 \iff b= -\frac{a+c}{2}$$$$ a^2+b^2+c^2=3 \iff 5b^2-2ac=3 $$$$ abc \leq 1 \iff 5b^3-3b-2 \leq 0 \iff b \leq 1 \iff |a+c| \leq 2 $$because : $$ \frac{5}{4}(a+c)^2 \leq 3+ \frac{1}{2}(a+c)^2 \iff |a+c| \leq 2 $$The other inequalities can be proved in a similar way. :-D
equality case : $ (a,b,c)=(-1,1,-1)$
The first one is trivial by Am-Gm. In fact, we have:
$$3 = a^2+b^2+c^2 \geq 3\sqrt[3]{a^2b^2c^2}$$Therefore
$$abc \leq |abc| \leq 1$$
This post has been edited 1 time. Last edited by GeoMorocco, Apr 19, 2025, 11:05 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SunnyEvan
129 posts
#7
Y by
GeoMorocco wrote:
SunnyEvan wrote:
$$ a+2b+c=0 \iff b= -\frac{a+c}{2}$$$$ a^2+b^2+c^2=3 \iff 5b^2-2ac=3 $$$$ abc \leq 1 \iff 5b^3-3b-2 \leq 0 \iff b \leq 1 \iff |a+c| \leq 2 $$because : $$ \frac{5}{4}(a+c)^2 \leq 3+ \frac{1}{2}(a+c)^2 \iff |a+c| \leq 2 $$The other inequalities can be proved in a similar way. :-D
equality case : $ (a,b,c)=(-1,1,-1)$
The first one is trivial by Am-Gm. In fact, we have:
$$3 = a^2+b^2+c^2 \geq 3\sqrt[3]{a^2b^2c^2}$$Therefore
$$abc \leq |abc| \leq 1$$

Yeah . :-D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42357 posts
#8
Y by
Thank you.
Z K Y
N Quick Reply
G
H
=
a