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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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f(xf(y)) + f(f(x) + f(y)) = yf(x) + f(x + f(y))
FelixD   12
N a minute ago by Bardia7003
Source: MEMO 2009, problem 1, single competition
Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that
\[ f(xf(y)) + f(f(x) + f(y)) = yf(x) + f(x + f(y))\]
holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.
12 replies
FelixD
Oct 1, 2009
Bardia7003
a minute ago
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Good chess board
sevket12   1
N 3 minutes ago by Turkish_sniper
Source: 2025 Turkey EGMO TST P1
A chessboard with some unit squares marked is called a $\textit{good board}$ if for any pair of rows \((s, t)\), a rook placed on a marked square in row \(s\) can reach a marked square in row \(t\) in several moves by only moving to marked squares above, below, or to the right of its current position.

Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete \(k\) columns and rearrange the remaining columns to form a $\textit{good board}$ determine the maximum possible value of \(k\).
1 reply
+1 w
sevket12
Feb 8, 2025
Turkish_sniper
3 minutes ago
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Function from the plane to the real numbers
AndreiVila   0
12 minutes ago
Source: Balkan MO Shortlist 2024 G7
Let $f:\pi\rightarrow\mathbb{R}$ be a function from the Euclidean plane to the real numbers such that $$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$$for any acute triangle $ABC$ with circumcenter $O$, centroid $G$ and orthocenter $H$. Prove that $f$ is constant.
0 replies
+1 w
AndreiVila
12 minutes ago
0 replies
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Fixed point in a small configuration
Assassino9931   2
N 17 minutes ago by Tamam
Source: Balkan MO Shortlist 2024 G3
Let $A, B, C, D$ be fixed points on this order on a line. Let $\omega$ be a variable circle through $C$ and $D$ and suppose it meets the perpendicular bisector of $CD$ at the points $X$ and $Y$. Let $Z$ and $T$ be the other points of intersection of $AX$ and $BY$ with $\omega$. Prove that $ZT$ passes through a fixed point independent of $\omega$.
2 replies
Assassino9931
Yesterday at 10:23 PM
Tamam
17 minutes ago
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Pair of multiples
Jalil_Huseynov   63
N 19 minutes ago by NerdyNashville
Source: APMO 2022 P1
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
63 replies
Jalil_Huseynov
May 17, 2022
NerdyNashville
19 minutes ago
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Geometric inequality with Fermat point
Assassino9931   3
N 21 minutes ago by mathuz
Source: Balkan MO Shortlist 2024 G2
Let $ABC$ be an acute triangle and let $P$ be an interior point for it such that $\angle APB = \angle BPC = \angle CPA$. Prove that
$$ \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}} \leq \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma}. $$When does equality hold?
3 replies
Assassino9931
Yesterday at 10:21 PM
mathuz
21 minutes ago
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Inequalities
hn111009   1
N 23 minutes ago by lbh_qys
Source: Somewhere
Let $a,b,c$ be non-negative number. Prove that $$\left(a+bc\right)^2+\left(b+ca\right)^2+\left(c+ab\right)^2\ge \sqrt{2}\left(a+b\right)\left(b+c\right)\left(c+a\right)$$
1 reply
hn111009
Yesterday at 3:29 PM
lbh_qys
23 minutes ago
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Geometry from Iranian TST 2017
bgn   17
N 33 minutes ago by SimplisticFormulas
Source: Iranian TST 2017, first exam, day1, problem 3
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.
Prove that $X,Y,Z$ are collinear.

Proposed by Hooman Fattahi
17 replies
bgn
Apr 5, 2017
SimplisticFormulas
33 minutes ago
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JBMO Shortlist 2021 G4
Lukaluce   1
N an hour ago by s27_SaparbekovUmar
Source: JBMO Shortlist 2021
Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.
1 reply
Lukaluce
Jul 2, 2022
s27_SaparbekovUmar
an hour ago
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incircle with center I of triangle ABC touches the side BC
orl   40
N 3 hours ago by Ilikeminecraft
Source: Vietnam TST 2003 for the 44th IMO, problem 2
Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)
40 replies
orl
Jun 26, 2005
Ilikeminecraft
3 hours ago
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amazing balkan combi
egxa   2
N 3 hours ago by ja.
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
2 replies
egxa
Yesterday at 1:57 PM
ja.
3 hours ago
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connected set in grid
David-Vieta   5
N 3 hours ago by zmm
Source: China High School Mathematics Olympics 2024 A P3
Given a positive integer $n$. Consider a $3 \times n$ grid, a set $S$ of squares is called connected if for any points $A \neq B$ in $S$, there exists an integer $l \ge 2$ and $l$ squares $A=C_1,C_2,\dots ,C_l=B$ in $S$ such that $C_i$ and $C_{i+1}$ shares a common side ($i=1,2,\dots,l-1$).

Find the largest integer $K$ satisfying that however the squares are colored black or white, there always exists a connected set $S$ for which the absolute value of the difference between the number of black and white squares is at least $K$.
5 replies
David-Vieta
Sep 8, 2024
zmm
3 hours ago
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functional equation interesting
skellyrah   10
N 4 hours ago by jasperE3
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$$
10 replies
skellyrah
Apr 24, 2025
jasperE3
4 hours ago
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Line through orthocenter
juckter   14
N 4 hours ago by lpieleanu
Source: Mexico National Olympiad 2011 Problem 2
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.
14 replies
juckter
Jun 22, 2014
lpieleanu
4 hours ago
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An upper bound for Iran TST 1996
Nguyenhuyen_AG   1
N Apr 14, 2025 by Victoria_Discalceata1
Let $a, \ b, \ c$ be the side lengths of a triangle. Prove that
\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]
1 reply
Nguyenhuyen_AG
Apr 13, 2025
Victoria_Discalceata1
Apr 14, 2025
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An upper bound for Iran TST 1996
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Reveal topic tags
inequalities
inequalities proposed
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Nguyenhuyen_AG
3316 posts
Nguyenhuyen_AG
#1 h  Apr 13, 2025, 8:55 AM
Y by
Let $a, \ b, \ c$ be the side lengths of a triangle. Prove that
\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]
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Victoria_Discalceata1
746 posts
Victoria_Discalceata1
#2 h  Apr 14, 2025, 2:33 PM
Y by
It holds strictly for equilateral triangles, because it reduces to $\frac{1}{9}\ge 0$. Now consider non-equilateral and non-degenerate triangles.
In terms of $Rrs$ it is $$16s^6-4r(56R-143r)s^4-8r^2\left(46R^2-206Rr-65r^2\right)s^2-36r^3(4R+r)^3\ge 0$$or $$\frac{AQ^3+Bq^3+CQ^2q+DQq^2}{(Q+q)^3}\ge 0,$$where $$\begin{aligned}A=16r^4\left(9207E^2+31936Er+27648r^2\right)&>0\\B=16\left(64E^6+736E^5r+3888E^4r^2+13016E^3r^3+30015E^2r^4+42944Er^5+27648r^6\right)&>0\\C= 16r^2\left(1188E^4+9568E^3r+47009E^2r^2+106816Er^3+82944r^4\right)&>0\\D=16r\left(544E^5+4884E^4r+22520E^3r^2+67817E^2r^3+117824Er^4+82944r^5\right)&>0\\E=R-2r&>0\\Q=4R^2+4Rr+3r^2-s^2&>0\\q=s^2-16Rr+5r^2&>0\end{aligned}$$Finally, consider the degenerate case.
If WLOG $a=b+c$ (including $c=0$) then the original inequality becomes $$\frac{(b-c)^2\left(16b^4+72b^3c+121b^2c^2+72bc^3+16c^4\right)}{36(b+c)^2(2b+c)^2(b+2c)^2}\ge 0$$
This post has been edited 1 time. Last edited by Victoria_Discalceata1, Apr 16, 2025, 1:31 AM
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