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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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Easy Combinatorics
MuradSafarli   1
N 15 minutes ago by Nuran2010
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{25}+2024)$ on the board?
1 reply
MuradSafarli
25 minutes ago
Nuran2010
15 minutes ago
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Sum of products is n mod 2
y-is-the-best-_   40
N 18 minutes ago by john0512
Source: IMO 2019 SL A5
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]
40 replies
y-is-the-best-_
Sep 22, 2020
john0512
18 minutes ago
J
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functional equation interesting
skellyrah   11
N 40 minutes ago by mkultra42
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$$
11 replies
skellyrah
Apr 24, 2025
mkultra42
40 minutes ago
J
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Perpendicularity
April   31
N an hour ago by zuat.e
Source: CGMO 2007 P5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
31 replies
April
Dec 28, 2008
zuat.e
an hour ago
J
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Functional equation on the set of reals
abeker   26
N an hour ago by Bardia7003
Source: MEMO 2017 I1
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$$f(x^2 + f(x)f(y)) = xf(x + y)$$for all real numbers $x$ and $y$.
26 replies
abeker
Aug 25, 2017
Bardia7003
an hour ago
J
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prefix sum QRs
optimusprime154   2
N an hour ago by GreekIdiot
Source: BMO 2025 P1
oops.....
2 replies
optimusprime154
2 hours ago
GreekIdiot
an hour ago
J
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Geometry with orthocenter config
thdnder   1
N 2 hours ago by thdnder
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
1 reply
thdnder
2 hours ago
thdnder
2 hours ago
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n = a*b , numbers of the form a^b
falantrng   3
N 2 hours ago by MuradSafarli
Source: Azerbaijan NMO 2023. Senior P1
The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
3 replies
falantrng
Aug 24, 2023
MuradSafarli
2 hours ago
J
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Inequality with 3 variables and a special condition
Nuran2010   1
N 2 hours ago by arqady
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
1 reply
Nuran2010
2 hours ago
arqady
2 hours ago
J
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find all functions
DNCT1   4
N 2 hours ago by jasperE3
Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that
$$f(2f(x)+2y)=f(2x+y)+y\quad\forall x,y,\in\mathbb{R^+} $$
4 replies
DNCT1
Oct 10, 2020
jasperE3
2 hours ago
J
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Reflection of D moves on a line
Ankoganit   4
N 2 hours ago by bin_sherlo
Source: KöMaL A. 705
Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

Michael Ren
4 replies
Ankoganit
Nov 11, 2017
bin_sherlo
2 hours ago
J
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USAMO 2003 Problem 1
MithsApprentice   67
N 2 hours ago by L13832
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
67 replies
MithsApprentice
Sep 27, 2005
L13832
2 hours ago
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D1024 : Can you do that?
Dattier   0
2 hours ago
Source: les dattes à Dattier
Let $x_{n+1}=x_n^2+1$ and $x_0=1$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
0 replies
Dattier
2 hours ago
0 replies
J
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Azer and Babek playing a game on a chessboard
Nuran2010   0
2 hours ago
Source: Azerbaijan Al-Khwarizmi IJMO TST
Azer and Babek have a $8 \times 8$ chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes $2$ rows and $2$ columns and looks at the $4$ cells in the intersection. Babek wants to have all these $4$ cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?
0 replies
Nuran2010
2 hours ago
0 replies
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J
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interesting ineq
nikiiiita   5
N Apr 6, 2025 by nikiiiita
Source: Own
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
5 replies
nikiiiita
Jan 29, 2025
nikiiiita
Apr 6, 2025
J
interesting ineq
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Reveal topic tags
inequalities
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Source: Own
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nikiiiita
100 posts
nikiiiita
#1 h  Jan 29, 2025, 10:27 AM • 1 Y
Y by PikaPika999
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
This post has been edited 1 time. Last edited by nikiiiita, Jan 29, 2025, 10:29 AM
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arqady
30218 posts
arqady
#3 h  Jan 30, 2025, 8:34 AM • 2 Y
Y by nikiiiita, PikaPika999
nikiiiita wrote:
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
By C-S $$\sum_{cyc}\sqrt{5a^2+2bc+2b}\leq\sqrt{\sum_{cyc}\frac{5a^2+2bc+2b}{2a+b+c}\sum_{cyc}(2a+b+c)}$$and it's enough to prove that:
$$\sum_{cyc}\frac{5a^2+2bc+2b}{2a+b+c}\leq\frac{27}{4}$$or
$$\sum_{cyc}(38a^3+141a^2b+149a^2c+120abc)\geq\sum_{cyc}(20a^4+76a^3b+76a^3c+120a^2b^2+156a^2bc)$$and after using AM-GM it's enough to prove $$\sum_{cyc}(38a^3+141a^2b+141a^2c+128abc)\geq\sum_{cyc}(20a^4+76a^3b+76a^3c+120a^2b^2+156a^2bc),$$which is smooth.
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nikiiiita
100 posts
nikiiiita
#4 h  Jan 30, 2025, 12:50 PM • 1 Y
Y by PikaPika999
Nice, thanks
For $a,b,c$ are non-negative reals, $a+b+c+abc=4$. Prove that:
$$\sqrt{ab+c^{2}+a}+\sqrt{bc+a^{2}+b}+\sqrt{ac+b^{2}+c} \ge 3\sqrt{\left(a+b+c\right)} $$Equality sign occurs for: $a=b=c=1$ or $a=b=0,c=4$ and its permutations
Is it old?
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arqady
30218 posts
arqady
#5 h  Jan 30, 2025, 4:27 PM • 1 Y
Y by PikaPika999
nikiiiita wrote:
For $a,b,c$ are non-negative reals, $a+b+c+abc=4$. Prove that:
$$\sqrt{ab+c^{2}+a}+\sqrt{bc+a^{2}+b}+\sqrt{ac+b^{2}+c} \ge 3\sqrt{\left(a+b+c\right)} $$Equality sign occurs for: $a=b=c=1$ or $a=b=0,c=4$ and its permutations
Is it old?
It's new for me. I think, it's very difficult to find a nice solution.
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nikiiiita
100 posts
nikiiiita
#6 h  Feb 13, 2025, 12:32 PM • 1 Y
Y by PikaPika999
bump for #4
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nikiiiita
100 posts
nikiiiita
#7 h  Apr 6, 2025, 8:58 PM • 1 Y
Y by PikaPika999
Any ideas?
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