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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
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 inequalities
sqing   2
N 2 minutes ago by lbh_qys
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
2 replies
1 viewing
sqing
20 minutes ago
lbh_qys
2 minutes ago
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Set: {f(r,r):r in S}=S
Sayan   7
N 4 minutes ago by kamatadu
Source: ISI (BS) 2007 #6
Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that
(i) $f(s,r)=f(r,s)$ for all $r,s \in S$
(ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$

Show that $\{f(r,r): r\in S\}=S$
7 replies
Sayan
Apr 11, 2012
kamatadu
4 minutes ago
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26 or 30 coins in a circle
NO_SQUARES   0
7 minutes ago
Source: Kvant 2025 no. 2 M2833
There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
Proposed by A. Gribalko
0 replies
NO_SQUARES
7 minutes ago
0 replies
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f(x,y)=0 iff (x,y) \in S, where |S|=2024
NO_SQUARES   0
14 minutes ago
Source: Kvant 2025 no. 2 M2832
There are $2024$ points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables $f(x,y)$ a) of degree $2025$; b) of degree $2024$ such that it equals to zero exactly at these marked points?
Proposed by Navid Safaei
0 replies
NO_SQUARES
14 minutes ago
0 replies
J
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Tangents forms triangle with two times less area
NO_SQUARES   0
17 minutes ago
Source: Kvant 2025 no. 2 M2831
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
0 replies
NO_SQUARES
17 minutes ago
0 replies
J
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IMO ShortList 2002, number theory problem 1
orl   76
N 24 minutes ago by NerdyNashville
Source: IMO ShortList 2002, number theory problem 1
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]
76 replies
orl
Sep 28, 2004
NerdyNashville
24 minutes ago
J
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Number of lucky numbers
NO_SQUARES   0
25 minutes ago
Source: Kvant 2025 no. 2 M2830
There are coins in denominations of $a$ and $b$ doubloons, where $a$ and $b$ are given mutually prime natural numbers, with $a < b < 100$. A non-negative integer $n$ is called lucky if the sum in $n$ doubloons can be scored with using no more than $1000$ coins. Find the number of lucky numbers.
From the folklore
0 replies
NO_SQUARES
25 minutes ago
0 replies
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Woaah a lot of external tangents
egxa   3
N an hour ago by NO_SQUARES
Source: All Russian 2025 11.7
A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point.
3 replies
egxa
Apr 18, 2025
NO_SQUARES
an hour ago
J
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2023 Hong Kong TST 3 (CHKMO) Problem 4
PikaNiko   3
N an hour ago by lightsynth123
Source: 2023 Hong Kong TST 3 (CHKMO)
Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$. Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$. Prove that the tangent at $E$ to $\Gamma$, the line $AD$ and the line $CN$ are concurrent.
3 replies
PikaNiko
Dec 3, 2022
lightsynth123
an hour ago
J
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Continuity of function and line segment of integer length
egxa   2
N an hour ago by NO_SQUARES
Source: All Russian 2025 11.8
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).
2 replies
egxa
Apr 18, 2025
NO_SQUARES
an hour ago
J
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Disjoint Pairs
MithsApprentice   41
N an hour ago by NerdyNashville
Source: USAMO 1998
Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.
41 replies
MithsApprentice
Oct 9, 2005
NerdyNashville
an hour ago
J
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Help my diagram has too many points
MarkBcc168   27
N 2 hours ago by Om245
Source: IMO Shortlist 2023 G6
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$.

Prove that $\angle BXP = \angle CXQ$.

Kian Moshiri, United Kingdom
27 replies
MarkBcc168
Jul 17, 2024
Om245
2 hours ago
J
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Geometry, SMO 2016, not easy
Zoom   18
N 3 hours ago by SimplisticFormulas
Source: Serbia National Olympiad 2016, day 1, P3
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.
18 replies
Zoom
Apr 1, 2016
SimplisticFormulas
3 hours ago
J
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A touching question on perpendicular lines
Tintarn   2
N 3 hours ago by pi_quadrat_sechstel
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 3
Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$.

The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$.

Show that the lines $AE$ and $CD$ are perpendicular.
2 replies
Tintarn
Mar 17, 2025
pi_quadrat_sechstel
3 hours ago
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J
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Find the constant
JK1603JK   1
N Apr 6, 2025 by Quantum-Phantom
Source: unknown
Find all $k$ such that $$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$forall $a,b,c\ge 0.$
1 reply
JK1603JK
Apr 6, 2025
Quantum-Phantom
Apr 6, 2025
J
Find the constant
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Reveal topic tags
inequalities
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JK1603JK
49 posts
JK1603JK
#1 h  Apr 6, 2025, 7:21 AM
Y by
Find all $k$ such that $$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$forall $a,b,c\ge 0.$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Quantum-Phantom
267 posts
Quantum-Phantom
#2 h  Apr 6, 2025, 10:47 AM • 1 Y
Y by teomihai
It suffices to find $k_{\max}$. Let $a=0$, $b=1$, then we get
\[k\le\frac{2c^4+c^3-2c^2+c+2}{2c^3-4c^2+2c}=:f(c),\]so $k_{\max}\le\min_{c>0}f(c)=\tfrac{9+4\sqrt2}2$. We prove the inequality for $k=\tfrac{9+4\sqrt2}2$. Let
\[u=\sum_{\rm cyc}a(a-b)(a-c),~v=\sum_{\rm cyc}a(b-c)^2,~w=\frac{(a+b+c)^3}{abc}\prod_{\rm cyc}(a-b)^2,\]then we have
\[\prod_{\rm cyc}(a-b)^2=\frac{uv^2w}{4u^3+12u^2v+12uv^2+4v^3+uw+4vw}.\]The desired inequality can be written as
\[(u+v)^2-u^2\ge\left(9-4\sqrt2\right)\prod_{\rm cyc}(a-b)^2,\]or
\[\frac{4v(u+v)^3(2u+v)+2vw\left(u-\sqrt2v\right)^2}{4(u+v)^3+(u+4v)w}\ge0.\]
This post has been edited 1 time. Last edited by Quantum-Phantom, Apr 6, 2025, 10:47 AM
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