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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.

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0 replies
jlacosta
May 1, 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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interesting geo config (2/3)
Royal_mhyasd   1
N an hour ago by Royal_mhyasd
Source: own
Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$. Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.
1 reply
Royal_mhyasd
an hour ago
Royal_mhyasd
an hour ago
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interesting geo config (1\3)
Royal_mhyasd   0
an hour ago
Source: own
Let $\triangle ABC$ be an acute triangle with $AC > AB$, $H$ its orthocenter and $O$ it's circumcenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = \angle ABC - \angle ACB$ and $P$ and $C$ are on different sides of $AB$. Denote by $S$ the intersection of the circumcircle of $\triangle ABC$ and $PA'$, where $A'$ is the reflection of $H$ over $BC$, $M$ the midpoint of $PH$, $Q$ the intersection of $OA$ and the parallel through $M$ to $AS$, $R$ the intersection of $MS$ and the perpendicular through $O$ to $PS$ and $N$ a point on $AS$ such that $NT \parallel PS$, where $T$ is the midpoint of $HS$. Prove that $Q, N, R$ lie on a line.

fiy it's 2am and i'm bored so i decided to look further into this interesting config that i had already made some observations on, maybe this problem is trivial from some theorem so if that's the case then i'm sorry lol :P i'll probably post 2 more problems related to it soon, i'd say they're easier than this though
0 replies
Royal_mhyasd
an hour ago
0 replies
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Parallel lines..
ts0_9   9
N an hour ago by OutKast
Source: Kazakhstan National Olympiad 2014 P3 D1 10 grade
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.
9 replies
ts0_9
Mar 26, 2014
OutKast
an hour ago
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KMN and PQR are tangent at a fixed point
hal9v4ik   4
N an hour ago by OutKast
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
4 replies
hal9v4ik
Mar 19, 2013
OutKast
an hour ago
J
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one cyclic formed by two cyclic
CrazyInMath   40
N 2 hours ago by HamstPan38825
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
40 replies
CrazyInMath
Apr 13, 2025
HamstPan38825
2 hours ago
J
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geometry problem
Medjl   5
N 2 hours ago by LeYohan
Source: Netherlands TST for IMO 2017 day 3 problem 1
A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ in $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$, at $P$. The point $L$ lies on $\omega$, and is such that $PL$ is tangent to $\omega$ and $L\neq Q$.
Show that $K, L$, and $M$ are collinear.
5 replies
Medjl
Feb 1, 2018
LeYohan
2 hours ago
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Connected, not n-colourable graph
mavropnevma   7
N 2 hours ago by OutKast
Source: Tuymaada 2013, Day 1, Problem 4 Juniors and 3 Seniors
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.

V. Dolnikov

EDIT. It is confirmed by the official solution that the graph is tacitly assumed to be finite.
7 replies
mavropnevma
Jul 20, 2013
OutKast
2 hours ago
J
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Homothety with incenter and circumcenters
Ikeronalio   8
N 2 hours ago by LeYohan
Source: Korea National Olympiad 2009 Problem 1
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $ BIC, CIA, AIB$. Let $ P, Q, R$ be the midpoints of segments $ DI, EI, FI $. Prove that the circumcenter of triangle $PQR $, $M$, is the midpoint of segment $IO$.
8 replies
Ikeronalio
Sep 9, 2012
LeYohan
2 hours ago
J
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2-var inequality
sqing   11
N 2 hours ago by ytChen
Source: Own
Let $ a,b>0 , a^2+b^2-ab\leq 1 . $ Prove that
$$a^3+b^3 -\frac{a^4}{b+1} -\frac{b^4}{a+1} \leq 1 $$
11 replies
sqing
May 27, 2025
ytChen
2 hours ago
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Sums of products of entries in a matrix
Stear14   0
2 hours ago
(a) $\ $Each entry of an $\ 8\times 8\ $ matrix equals either $\ 1\ $ or $\ 2.\ $ Let $\ A\ $ denote the sum of eight products of entries in each row. Also, let $\ B\ $ denote the sum of eight products of entries in each column. Find the maximum possible value of $\ A-B.\ $ In other words, find
$$ {\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} - \sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right] $$
(b) $\ $Same question, but for a $\ 2025\times 2025\ $ matrix.
0 replies
Stear14
2 hours ago
0 replies
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a father and his son are skating around a circular skating rink
parmenides51   2
N 3 hours ago by thespacebar1729
Source: Tournament Of Towns Spring 1999 Junior 0 Level p1
A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?

(Tairova)
2 replies
parmenides51
May 7, 2020
thespacebar1729
3 hours ago
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Sums of n mod k
EthanWYX2009   1
N 3 hours ago by Martin.s
Source: 2025 May 谜之竞赛-3
Given $0<\varepsilon <1.$ Show that there exists a constant $c>0,$ such that for all positive integer $n,$
\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]Proposed by Cheng Jiang
1 reply
1 viewing
EthanWYX2009
May 26, 2025
Martin.s
3 hours ago
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Easy P4 combi game with nt flavour
Maths_VC   1
N 5 hours ago by p.lazarov06
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
1 reply
Maths_VC
May 27, 2025
p.lazarov06
5 hours ago
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Central sequences
EeEeRUT   14
N 6 hours ago by HamstPan38825
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
14 replies
EeEeRUT
Apr 16, 2025
HamstPan38825
6 hours ago
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J
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help me please!
iptgut   6
N Sep 25, 2009 by leedt26
6 replies
iptgut
Sep 24, 2009
leedt26
Sep 25, 2009
J
help me please!
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Reveal topic tags
logarithms
inequalities unsolved
inequalities
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iptgut
51 posts
iptgut
#1 h  Sep 24, 2009, 12:40 PM • 2 Y
Y by Adventure10, Mango247
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FelixD
588 posts
FelixD
#2 h  Sep 24, 2009, 1:11 PM • 1 Y
Y by Adventure10
Substitute $ a = x + y$ and so on. Then you get
\[ \sum \frac {x + y}{x + 3y} \ge \frac {3}{2}\]
or
\[ \frac {3}{4} \ge \sum \frac {1}{e^a + 3},\]
where $ \frac {y}{z} = e^a$ and so on. Hence $ a + b + c = 0$. Then you have to distinguish 3 cases ($ a,b,c \le \ln 3$, $ a,b \le \ln 3$, $ a \le \ln 3$, where $ a \le b \le c$). Every case can be done by Karamata, noting that $ f(t) = \frac {1}{3 + e^t}$ is concave on $ [ - \infty, \ln 3]$ and convex on $ [\ln 3, \infty]$ and Am-Gm in the end.

I used $ a, b, c$ twice, but they are not the same…^^ I just recognised that after writing the post...
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kuing
1008 posts
kuing
#3 h  Sep 24, 2009, 1:36 PM • 1 Y
Y by Adventure10
we can setting $ a = x + y,b = y + z,c = z + x,x,y,z > 0$, that we have\[ \sum {\frac {a}{{2a + b - c}}} = \sum {\frac {{x + y}}{{x + 3y}}} = 3 - 2\sum {\frac {y}{{x + 3y}}}\]so\[ \sum {\frac {a}{{2a + b - c}}} \ge \frac {3}{2}\]\[ \Leftrightarrow 3 - 2\sum {\frac {y}{{x + 3y}}} \ge \frac {3}{2}\]\[ \Leftrightarrow \sum {\frac {1}{{\frac {x}{y} + 3}}} \le \frac {3}{4}\]let $ \frac {x}{y} = t,\frac {y}{z} = u,\frac {z}{x} = v$, that we have$ t,u,v > 0$ and $ tuv = 1$, just need to prove that\[ \sum {\frac {1}{{t + 3}}} \le \frac {3}{4}\]\[ \Leftrightarrow 3\left( {t + u + v} \right) + 5\left( {tu + uv + vt} \right) - 24 \ge 0\]which is ture by Am-Gm.
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iptgut
51 posts
iptgut
#4 h  Sep 25, 2009, 7:16 AM • 2 Y
Y by Adventure10, Mango247
Thank you!
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kuing
1008 posts
kuing
#5 h  Sep 25, 2009, 10:22 AM • 1 Y
Y by Adventure10
test

how to delete my post... :maybe:
This post has been edited 4 times. Last edited by kuing, Sep 25, 2009, 10:42 AM
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FelixD
588 posts
FelixD
#6 h  Sep 25, 2009, 10:27 AM • 1 Y
Y by Adventure10
What does this mean, kuing? :)
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leedt26
158 posts
leedt26
#7 h  Sep 25, 2009, 11:06 AM • 2 Y
Y by Adventure10, Mango247
It's posted in here: http://www.mathlinks.ro/viewtopic.php?p=1559251#1559251
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