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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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:
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0 replies
jlacosta
Jun 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
J
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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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Inequality
eric201291   0
7 minutes ago
Real number a,b,c, a+b+c=3, Prove:(a^2+1)(b^2+1)(c^2+1)≥(a+1)(b+1)(c+1).
0 replies
eric201291
7 minutes ago
0 replies
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Inequalities
sqing   5
N 12 minutes ago by sqing
Let $ 0\leq a \leq \frac{1}{2}$. Prove that
$$ \sqrt{\frac{2}{3}a+a^3}+\sqrt{\frac{2}{3}a+(1-2a)^3}+\sqrt{\frac{2}{3}(1-2a)+a^3} \geq \sqrt{\frac{7}{3}}$$$$ \sqrt{\frac{3}{4}a+a^3}+\sqrt{\frac{3}{4}a+(1-2a)^3}+\sqrt{\frac{3}{4}(1-2a)+a^3} \geq \frac{1}{2}\sqrt{\frac{31}{3}}$$$$\sqrt{\frac{11}{8}a+a^3}+\sqrt{\frac{11}{8}a+(1-2a)^3}+\sqrt{\frac{11}{8}(1-2a)+a^3} \geq \frac{\sqrt{2}+ \sqrt{11}+\sqrt{13}}{4}$$$$ \sqrt{\frac{31}{20}a+a^3}+\sqrt{\frac{31}{20}a+(1-2a)^3}+\sqrt{\frac{31}{20}(1-2a)+a^3} \geq \frac{5+6\sqrt{5}+\sqrt{155}}{10\sqrt{2}}$$
5 replies
sqing
Tuesday at 2:45 PM
sqing
12 minutes ago
J
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Inequalities from SXTB
sqing   27
N 15 minutes ago by sqing
T2811. Let $ a,b,c,d> 0 $ and $ a+b+c+d=1. $ Prove that$\frac{a}{a+1}+\frac{2b}{b+2}+\frac{3c}{c+3}+\frac{6d}{d+6} \leq \frac{12}{13}$
27 replies
sqing
Dec 5, 2024
sqing
15 minutes ago
J
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Tricky FE
Rijul saini   6
N 21 minutes ago by megarnie
Source: LMAO 2025 Day 1 Problem 1
Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$f(xy) + f(f(y)) = f((x + 1)f(y))$$for all real numbers $x$, $y$.

Proposed by MV Adhitya and Kanav Talwar
6 replies
Rijul saini
Yesterday at 6:58 PM
megarnie
21 minutes ago
J
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2-var inequality
sqing   3
N 22 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $\frac{1}{a^2+3} + \frac{1}{b^2+3} -ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{3(\sqrt{57}-7)}{4}$$Let $ a,b\geq 0 $ and $\frac{a}{b^2+3} + \frac{b}{a^2+3} +ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \leq \frac{9}{4}$$Let $ a,b\geq 0 $ and $ \frac{a}{b^3+3}+\frac{b}{a^3+3}-ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{9}{4}$$
3 replies
sqing
Yesterday at 12:55 PM
sqing
22 minutes ago
J
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geometry with quadrilateral, tangent circles wanted
trying_to_solve_br   56
N 23 minutes ago by Adywastaken
Source: IMO 2020 Shortlist G3
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.

$\emph{Slovakia}$
56 replies
trying_to_solve_br
Jul 20, 2021
Adywastaken
23 minutes ago
J
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Minimization without derivative
Butterfly   3
N 25 minutes ago by nabodorbuco2

Find the minimum value of $f(x)=\frac{(3x+2)^4}{x(x+3)}$ on $(0,+\infty)$ without derivative.
3 replies
Butterfly
3 hours ago
nabodorbuco2
25 minutes ago
J
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Geometry
MathsII-enjoy   8
N 27 minutes ago by aaravdodhia
Given triangle $ABC$ inscribed in $(O)$, $M, N$ are respectively the midpoints of the major and minor arcs $BC$. Let $I$ be the center of the inscribed circle, $R$ be $A-mix$, $D$ is the intersection point of the line through $A$ parallel to $BC$ with $(O)$. $DI$ intersects $AR$ at $K$, take $L$ on $AK$ so that $LI//BC$. $NL$ intersects $(O)$ at $G$, $AG$ intersects $LI$ at $Z$. Prove that: $ZM$ is perpendicular to $KN$.
8 replies
MathsII-enjoy
May 23, 2025
aaravdodhia
27 minutes ago
J
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11th PMO Nationals, Easy #5
scarlet128   0
27 minutes ago
Source: https://pmo.ph/wp-content/uploads/2020/12/11th-PMO-Questions.pdf
Solve for x : 2(floor of x) = x + 2{x}
0 replies
scarlet128
27 minutes ago
0 replies
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Problem 2 IMO 2005 (Day 1)
Valentin Vornicu   82
N 38 minutes ago by mahyar_ais
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.

Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
82 replies
Valentin Vornicu
Jul 13, 2005
mahyar_ais
38 minutes ago
J
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Strange multi-equation
giangtruong13   1
N 39 minutes ago by iniffur
Source: One of Vietnamese Specialized School's Math Entrance Exam
Solve the multi equation: $\begin{cases} x^3+x^2-xy^2=\sqrt{(x-y^2)^3} \\ 56x^2+20(x^2-y^2)=\sqrt[3]{4x(8x+1)}-2\end{cases}$
1 reply
giangtruong13
Yesterday at 3:07 PM
iniffur
39 minutes ago
J
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One of the lines is tangent
Rijul saini   5
N an hour ago by AN1729
Source: LMAO 2025 Day 2 Problem 2
Let $ABC$ be a scalene triangle with incircle $\omega$. Denote by $N$ the midpoint of arc $BAC$ in the circumcircle of $ABC$, and by $D$ the point where the $A$-excircle touches $BC$. Suppose the circumcircle of $AND$ meets $BC$ again at $P \neq D$ and intersects $\omega$ at two points $X$, $Y$.

Prove that either $PX$ or $PY$ is tangent to $\omega$.

Proposed by Sanjana Philo Chacko
5 replies
Rijul saini
Yesterday at 7:02 PM
AN1729
an hour ago
J
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Sum_cyc sqrt {2014^2+7}<= sqrt {2014+7} (a+b+c)
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
Let $a,\ b,\ c$ be positive real numbers such that $abc=1.$

Prove that :

\[\sqrt{2014a^2+7}+\sqrt{2014b^2+7}+\sqrt{2014c^2+7}\leq \sqrt{2014+7}\ (a+b+c).\]


Proposed by Kunihiko Chikaya
1 reply
Kunihiko_Chikaya
Jul 6, 2014
Mathzeus1024
an hour ago
J
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IMO problem 1
iandrei   78
N an hour ago by maromex
Source: IMO ShortList 2003, combinatorics problem 1
Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.
78 replies
iandrei
Jul 14, 2003
maromex
an hour ago
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computation wih equiangular hexagon (2015 AESC MSU Internet Olympiad 7.4)
parmenides51   2
N Feb 18, 2021 by hi_im_bob
In convex hexagon $ABCDEF$, all interior angles are equal. It is known that $AB = 2$, $CD = 5$, $DE = 7$, $EF = 1$. Find $BC$ and $AF$.
2 replies
parmenides51
Feb 18, 2021
hi_im_bob
Feb 18, 2021
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computation wih equiangular hexagon (2015 AESC MSU Internet Olympiad 7.4)
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Reveal topic tags
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parmenides51
30653 posts
parmenides51
#1 h  Feb 18, 2021, 8:07 PM
Y by
In convex hexagon $ABCDEF$, all interior angles are equal. It is known that $AB = 2$, $CD = 5$, $DE = 7$, $EF = 1$. Find $BC$ and $AF$.
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donian9265
541 posts
donian9265
#2 h  Feb 18, 2021, 8:42 PM
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solution
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hi_im_bob
903 posts
hi_im_bob
#3 h  Feb 18, 2021, 9:12 PM • 4 Y
Y by ilovepizza2020, Helloitsaaryan, tenebrine, chaboyamathcounts
Whenever there's an equiangular hexagon, always extend sides to form an equilateral triangle. This will provide you with sufficient information to find the missing lengths.
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