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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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Turbo's en route to visit each cell of the board
Lukaluce   21
N 2 minutes ago by zRevenant
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
21 replies
Lukaluce
Apr 14, 2025
zRevenant
2 minutes ago
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\frac{1}{5-2a}
Havu   0
7 minutes ago
Let $a,b,c \ge \frac{1}{2}$ and $a^2+b^2+c^2=3$. Find minimum:
\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]
0 replies
Havu
7 minutes ago
0 replies
J
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interesting function equation (fe) in IR
skellyrah   0
12 minutes ago
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
0 replies
skellyrah
12 minutes ago
0 replies
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Classical looking graph
matinyousefi   4
N 17 minutes ago by Rohit-2006
Source: Iranian Our MO 2020 P4
In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes.

Proposed by Mohammad Moshtaghi Far, Kian Shamsaie Rated 4
4 replies
matinyousefi
Mar 11, 2020
Rohit-2006
17 minutes ago
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Checking a summand property for integers sufficiently large.
DinDean   3
N 32 minutes ago by DinDean
For any fixed integer $m\geqslant 2$, prove that there exists a positive integer $f(m)$, such that for any integer $n\geqslant f(m)$, $n$ can be expressed by a sum of positive integers $a_i$'s as
\[n=a_1+a_2+\dots+a_m,\]where $a_1\mid a_2$, $a_2\mid a_3$, $\dots$, $a_{m-1}\mid a_m$ and $1\leqslant a_1<a_2<\dots<a_m$.
3 replies
DinDean
Yesterday at 5:21 PM
DinDean
32 minutes ago
J
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Interesting inequalities
sqing   2
N 39 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
sqing
an hour ago
lbh_qys
39 minutes ago
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Set: {f(r,r):r in S}=S
Sayan   7
N 41 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
41 minutes ago
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26 or 30 coins in a circle
NO_SQUARES   0
an hour 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
an hour ago
0 replies
J
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f(x,y)=0 iff (x,y) \in S, where |S|=2024
NO_SQUARES   0
an hour 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
an hour ago
0 replies
J
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Tangents forms triangle with two times less area
NO_SQUARES   0
an hour 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
an hour ago
0 replies
J
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IMO ShortList 2002, number theory problem 1
orl   76
N an hour 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
an hour ago
J
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Number of lucky numbers
NO_SQUARES   0
an hour 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
an hour ago
0 replies
J
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Woaah a lot of external tangents
egxa   3
N 2 hours 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
2 hours ago
J
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2023 Hong Kong TST 3 (CHKMO) Problem 4
PikaNiko   3
N 2 hours 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
2 hours ago
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J
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The locus of P with supplementary angles condition
WakeUp   3
N Apr 4, 2025 by Nari_Tom
Source: Baltic Way 2001
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
3 replies
WakeUp
Nov 17, 2010
Nari_Tom
Apr 4, 2025
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The locus of P with supplementary angles condition
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Reveal topic tags
geometry
rhombus
parallelogram
circumcircle
conics
hyperbola
cyclic quadrilateral
L
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Source: Baltic Way 2001
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WakeUp
1347 posts
WakeUp
#1 h  Nov 17, 2010, 7:19 PM • 2 Y
Y by Adventure10, Mango247
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
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yetti
2643 posts
yetti
#2 h  Nov 17, 2010, 8:29 PM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be just a parallelogram and $\angle APD + \angle CPB = 180^\circ$ $\Longrightarrow$ circumcircles $(O_1), (O_3)$ of $\triangle APD, \triangle CPB$ are congruent. Translating $\triangle APD$ by $\overrightarrow{AB}$ into a $\triangle CP'B$ creates a cyclic quadrilateral $PBP'C$ and parallelogram $PABP'$ $\Longrightarrow$ circumcircles $(O_2), (O_3)$ of $\triangle BPA, \triangle CPB$ are also congruent. It follows that $\angle PCD = \angle DAP,$ which means that isogonal conjugate $P^*$ of $P$ WRT $\triangle ACD$ is on perpendicular bisector of $AC.$ As a result, the locus of $P$ is a rectangular circum-hyperbola of the parallelogram $ABCD,$ centered at its diagonal intersection $E.$ If $ABCD$ is a rhombus, this hyperbola degenerates to its diagonals $AC, BD.$
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isaacmeng
113 posts
isaacmeng
#3 h  May 5, 2020, 10:45 AM
Y by
What if P is not necessarily in the plane ABCD?
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Nari_Tom
116 posts
Nari_Tom
#4 h  Apr 4, 2025, 10:01 AM
Y by
There is a theorem states that if $P$ is a point lies in the plane of $ABCD$ and $\angle APD+\angle CPD=180^{\circ}$, then there exists a isogonal conjugate of $P$. Since $ABCD$ is a rhombus we can easily conclude that $P$ lies on $AC$ or $BD$.
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