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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
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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]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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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
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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
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2025 USC Math Comp (SCMC) individual round
Bluesoul   0
37 minutes ago
1. For an integer $x,$ we define a \textit{step} as either doubling the value of the integer or subtracting 3 from it. What is the minimum number of steps required to obtain 25 from 11?
2. Find the sum of all integer values of $n$ that satisfy the inequality chain \[n^3<2025<3^n.\]3. A rectangle with length 20 units and height 16 units is divided into 10 smaller congruent rectangles. Let $P$ be the largest possible perimeter of one of these small rectangles; compute the value of $10P.$
4. Positive integers $x,y,z$ satisfy the system
\[\left\{\begin{array}{l} x^2+y^2=z^2+22\\ y^2+z^2=x^2+76\\ x^2+z^2=y^2-4. \end{array} \right.\]What is the value of $xyz?$
5. \begin{problem}
Four husband-wife couples go ballroom dancing one evening. The husbands' names are Henry, Peter, Steve, and Roger, while the wives' names are Elizabeth, Keira, Mary, and Anne. At a given moment, Henry's wife is dancing with Elizabeth's husband, who is not Henry; Roger and Anne are not dancing; Peter is playing the trumpet; and Mary is playing the piano. Given that Anne's husband is not Peter, how many different letters are in the name of Roger's wife?
6. A laser is fired from vertex $A$ into the interior of regular hexagon $ABCDEF,$ whose sides are mirrors, and hits side $\overline{CD}$ at $G.$ It then reflects and hits $\overline{AF}$ at $H,$ and finally reflects and hits $\overline{DE}$ at $I.$ If $\angle BAG=45^\circ,$ then how many degrees are in $\angle HIE?$
7. What is the value of the expression
\[\frac{\log_2(\log_2 3)}{\log_4(\log_4 9)}?\]8. The area of equiangular octagon $ABCDEFGH,$ with $AB=EF=2,$ $BC=FG=3,$ $CD=GH=4,$ and $DE=AH=5,$ can be written in the form $a+b\sqrt{2}$, find $a+b$.
9. A toe-wrestling tournament between Don and Kam consists of three matches. In each match, the winner is the first person to reach five points. After the three matches, each person’s score is the number of matches they won, plus the sum of the points they earned during all of their matches. Let $d$ and $k$ denote Don and Kam’s final scores, respectively. How many ordered pairs $(d, k)$ are possible?
10. For each positive integer $n$, let $s(n)$ denote the sum of the remainders when $n$ is divided by $2,3,4,5,$ and $6.$ For example, when $n=93,$ we have $s(93)=1+0+1+3+3=8.$ Compute the integer $N$ for which \[\sum_{n=1}^{N}s(n)=2025.\]11. For complex numbers $z,$ we define the function \[f(z)=\frac{z+3}{z-2i}.\]Over all values of $z$ for which $f(z)$ is real, the minimum possible value of $|z|^2$ can be written in the form $\dfrac{m}{n}$ for positive integers $m$ and $n.$ Compute the value of $100m+n.$
12. In convex quadrilateral $ABCD,$ $AB=6, BC=10,$ and $\angle{ABC}=90^{\circ}$. Let $M$ and $N$ be the midpoints of $\overline{AD}$ and $\overline{CD},$ respectively. Compute the area of $\triangle{BMN},$ given that the area of $ABCD$ is $50$.
13. What is the remainder when $20^{25}$ is divided by 2025?
14. Your friend plays a prank on you by changing your phone's password. Your friend chooses a password consisting of 4 decimal digits $\overline{abcd}$ uniformly at random and tells you the sum of its digits. (Leading zeros are allowed, so your friend can choose any password from 0000, 0001, and so on to 9999.) Then, you select a digit $e;$ your friend tells you the password if and only if $e$ is the median of the set $\{a,b,c,d,e\}.$
\null
Now, your friend picks a password whose digits sum to 20; let $S$ be the set of all such passwords. Suppose you select $e$ such that the probability that your friend tells you the password, given this information, is maximized. Compute the number of passwords in $S$ for which this would not occur, given your choice of $e.$
15. For real numbers $x,$ we define the function \[f(x)=\lceil{1+\sqrt{x+1}}\rceil+\lfloor{1-\sqrt{x-1}}\rfloor.\]Compute the $100^\text{th}-$smallest integer $x$ for which $f(x)=2$.
16. How many ordered pairs $(x,y),$ with $1\le x,y\le 100,$ satisfy the congruence \[2^{2^x+2^y}\equiv 1\pmod{101}?\]17. $\triangle ABC$ has circumcircle $\omega$ and incenter $I.$ $\overline{AI}$ is extended to intersect $\omega$ at a point $P\ne A,$ and $\overline{BI}$ is extended to intersect $\overline{AC}$ at $Q.$ If $AB=5,$ $BC=8,$ and $IPCQ$ is a cyclic quadrilateral, then compute $AC^2.$
18. The $\textbf{Cantor set}$ is constructed as follows:
i) Start with the closed interval $[0, 1]$.
ii)Remove the open middle third of the interval, so we remove $\left(\frac{1}{3}, \frac{2}{3}\right)$ at first and leave $\left[0, \frac{1}{3}\right]$ and $\left[\frac{2}{3}, 1\right]$.
iii) Remove the open middle third from each of the remaining closed intervals, and repeat this step infinitely.
For how many integer values of $i$, where $0 \leq i \leq 10$, is $\frac{i}{10}$ an element of the Cantor set?
19. For complex numbers $a,b,c$ satisfying $|a|^2+|b|^2+|c|^2=1$, the maximum value of $|ab(a^2-b^2)+ca(c^2-a^2)+bc(b^2-c^2)|$ can be expressed in the simplest form of $\frac{p}{q}, \gcd(p,q)=1$, find $p+q$.
20. Consider cyclic quadrilateral $ABCD$ with all integer side lengths and $AB=AD=6$. Let $AC$ meet $BD$ at $F$, $AF=3,CF=9$. Denote the centers of the circumcircles of polygons $CBF, ABCD, DCF$ as $H,I,J$ respectively. Compute the area of $\triangle{HIJ}$. The answer is in the simplest form of $\frac{p\sqrt{q}}{r},\gcd(p,r)=1$ and $q$ is square-free, compute $p+q+r$.
0 replies
Bluesoul
37 minutes ago
0 replies
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Oi! These lines concur
Rg230403   21
N an hour ago by ohiorizzler1434
Source: LMAO 2021 P5, LMAOSL G3(simplified)
Let $I, O$ and $\Gamma$ respectively be the incentre, circumcentre and circumcircle of triangle $ABC$. Points $A_1, A_2$ are chosen on $\Gamma$, such that $AA_1 = AI = AA_2$, and point $A'$ is the foot of the altitude from $I$ to $A_1A_2$. If $B', C'$ are similarly defined, prove that lines $AA', BB'$ and $CC'$ concurr on $OI$.
Original Version from SL
Proposed by Mahavir Gandhi
21 replies
1 viewing
Rg230403
May 10, 2021
ohiorizzler1434
an hour ago
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Inequalities
sqing   10
N an hour ago by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
10 replies
sqing
Mar 26, 2025
sqing
an hour ago
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2024 BxMO P3
beansenthusiast505   4
N an hour ago by GeorgeMetrical123
Source: 2024 BxMO P3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $\left|AC\right|\neq\left|BC\right|$. The internal angle bisector of $\angle CAB$ intersects side $BC$ at $D$ and the external angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $\Omega$ at $E$ and $F$ respectively. Let $G$ be the intersection of lines $AE$ and $FI$ and let $\Gamma$ be the circumcircle of triangle $BDI$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.
4 replies
beansenthusiast505
Apr 28, 2024
GeorgeMetrical123
an hour ago
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Length Condition on Circumcenter Implies Tangency
ike.chen   41
N 2 hours ago by ravengsd
Source: ISL 2022/G4
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
41 replies
ike.chen
Jul 9, 2023
ravengsd
2 hours ago
J
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Calculating combinatorial numbers
lgx57   5
N 2 hours ago by generatingFraction
Try to simplify this expression:

$$\sum_{i=1}^n \sum_{j=1}^i C_{n}^i C_{n}^j$$
5 replies
lgx57
3 hours ago
generatingFraction
2 hours ago
J
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Probably appeared before
steven_zhang123   2
N 2 hours ago by whwlqkd
In the plane, there are two line segments $AB$ and $CD$, with $AB \neq CD$. Prove that there exists and only exists one point $P$ such that $\triangle PAB \sim \triangle PCD$.($P$ corresponds to $P$, $A$ corresponds to $C$)
Click to reveal hidden text
2 replies
steven_zhang123
Today at 2:29 AM
whwlqkd
2 hours ago
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Hard geometry
jannatiar   3
N 2 hours ago by alinazarboland
Source: 2024 AlborzMO P4
In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear.

Proposed by Ali Nazarboland
3 replies
jannatiar
Mar 4, 2025
alinazarboland
2 hours ago
J
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Proper subsets of R
lgx57   0
3 hours ago
Let $S_1,S_2 \cdots S_n$ are proper subsets of $\mathbb{R}$ and they are closed for addition and subtraction. Try to prove that:

$$\displaystyle\bigcup_{i=1}^n S_i \ne \mathbb{R}$$
0 replies
lgx57
3 hours ago
0 replies
J
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Proving ZA=ZB
nAalniaOMliO   4
N 3 hours ago by Primeniyazidayi
Source: Belarusian National Olympiad 2025
Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$.
Prove that $ZA=ZH$.
Vadzim Kamianetski
4 replies
nAalniaOMliO
Friday at 8:36 PM
Primeniyazidayi
3 hours ago
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IGO 2022 advanced/free P2
Tafi_ak   16
N 4 hours ago by mcmp
Source: Iranian Geometry Olympiad 2022 P2 Advanced, Free
We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.

Proposed by Patrik Bak, Slovakia
16 replies
Tafi_ak
Dec 13, 2022
mcmp
4 hours ago
J
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100 Points but the Contestants get 0
tastymath75025   29
N 5 hours ago by popop614
Source: USA Winter TST for IMO 2020, Problem 6, by Michael Ren
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.

Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.

Michael Ren
29 replies
tastymath75025
Jan 27, 2020
popop614
5 hours ago
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GOTEEM #5: Circumcircle passes through fixed point
tworigami   22
N 5 hours ago by ohiorizzler1434
Source: GOTEEM: Mock Geometry Contest
Let $ABC$ be a triangle and let $B_1$ and $C_1$ be variable points on sides $\overline{BA}$ and $\overline{CA}$, respectively, such that $BB_1 = CC_1$. Let $B_2 \neq B_1$ denote the point on $\odot(ACB_1)$ such that $BC_1$ is parallel to $B_1B_2$, and let $C_2 \neq C_1$ denote the point on $\odot(ABC_1)$ such that $CB_1$ is parallel to $C_1C_2$. Prove that as $B_1, C_1$ vary, the circumcircle of $\triangle AB_2C_2$ passes through a fixed point, other than $A$.

Proposed by tworigami
22 replies
tworigami
Jan 2, 2020
ohiorizzler1434
5 hours ago
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Easy Geometry
pokmui9909   3
N 6 hours ago by whwlqkd
Source: FKMO 2025 P4
Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.
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pokmui9909
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whwlqkd
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sqing
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#1 h  Mar 26, 2025, 3:07 AM
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Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
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sqing
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#2 h  Mar 26, 2025, 3:26 AM
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Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that
$$a+b+c\geq1$$$$a+b+2c\geq\frac{9}{8}$$$$a+b+1.664c\geq\frac{9}{8}$$$$a+b+\frac{7}{5}c\geq \frac{9}{8}\sqrt [3]{\frac{7}{5}}-\frac{7}{40}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+ ab +18abc\geq\frac{343}{324} $. Prove that
$$a+b+c\geq1$$$$a+b+\frac{23}{20}c\geq\frac{7\sqrt 7}{18}$$$$a+b+1.1409c \geq\frac{7\sqrt 7}{18}$$$$a+b+\frac{11}{10}c\geq \frac{7}{6}\sqrt [3]{\frac{11}{10}}-\frac{11}{60}$$
This post has been edited 4 times. Last edited by sqing, Mar 26, 2025, 4:10 AM
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DAVROS
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#3 h  Mar 26, 2025, 6:51 AM
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sqing wrote:
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that $a^2+b^2+ ab +21abc\leq\frac{512}{441}$
solution
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sqing
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sqing
#4 h  Mar 26, 2025, 7:03 AM
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Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$ a^2+b^2+c +13abc\leq\frac{2997+23\sqrt{69}}{3042}$$$$ a^2+b^2+c +14abc\leq\frac{11(121+2\sqrt{22})}{1323}$$
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sqing
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#5 h  Mar 26, 2025, 7:04 AM
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Very very nice.Thank DAVROS.
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sqing
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#6 h  Mar 26, 2025, 8:04 AM
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Let $ a,b,c\geq 0 $ and $ a^2+b^2+c +ab+10 abc\geq\frac{28}{27}$. Prove that
$$ a+b+\frac{8}{9}c \geq \frac{224}{243}$$$$ a+b+\frac{529}{500}c \geq \frac{2}{3}\sqrt{\frac{7}{3}}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+c+ ab +\frac{19}{2}abc\geq\frac{55}{54} $. Prove that
$$ a+b+c \geq 1$$$$ a+b+\frac{9}{10}c \geq \frac{11}{12}$$$$ a+b+\frac{1029}{1000}c \geq \frac{1}{3}\sqrt{\frac{55}{6}}$$
This post has been edited 1 time. Last edited by sqing, Mar 26, 2025, 8:17 AM
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DAVROS
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#7 h  Mar 26, 2025, 3:35 PM
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sqing wrote:
Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that $a+b+2c\geq\frac{9}{8}$
solution
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DAVROS
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#8 h  Mar 26, 2025, 4:17 PM
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sqing wrote:
Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that $a+b+\frac{7}{5}c\geq \frac{9}{8}\sqrt [3]{\frac{7}{5}}-\frac{7}{40}$
solution
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sqing
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#9 h  Mar 27, 2025, 2:42 AM
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Very very nice.Thank DAVROS.
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MathRook7817
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#10 h  Mar 27, 2025, 5:26 AM
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I swear these two are like the best duo.
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sqing
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#11 h  an hour ago
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Let $ a,b,c $ be reals such that $ a+b+c=0. $ Prove that
$$ \frac{a+b+2}{c^2+c+3} + \frac{a+c+2}{b^2+b+3} \leq 2$$$$\frac{a+b+2}{c^2+c+10} + \frac{a+c+2}{b^2+b+10} \leq \frac{2}{3}$$$$ \frac{a+b+1}{c^2+c+7} + \frac{a+c+1}{b^2+b+7} \leq \frac{2}{3}$$
This post has been edited 1 time. Last edited by sqing, an hour ago
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