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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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Continuity of function and line segment of integer length
egxa   2
N 4 minutes 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
4 minutes ago
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Disjoint Pairs
MithsApprentice   41
N 7 minutes 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
7 minutes ago
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Help my diagram has too many points
MarkBcc168   27
N 19 minutes 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
19 minutes ago
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Geometry, SMO 2016, not easy
Zoom   18
N an hour 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
an hour ago
J
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A touching question on perpendicular lines
Tintarn   2
N an hour 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
an hour ago
J
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Woaah a lot of external tangents
egxa   2
N 2 hours ago by soryn
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.
2 replies
egxa
Apr 18, 2025
soryn
2 hours ago
J
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Some nice summations
amitwa.exe   31
N 2 hours ago by soryn
Problem 1: $\Omega=\left(\sum_{0\le i\le j\le k}^{\infty} \frac{1}{3^i\cdot4^j\cdot5^k}\right)\left(\mathop{{\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}}}_{i\neq j\neq k}\frac{1}{3^i\cdot3^j\cdot3^k}\right)=?$
31 replies
amitwa.exe
May 24, 2024
soryn
2 hours ago
J
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Interesting inequalities
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 ,b+c-ca=1 $ and $ c+a-ab=3.$ Prove that
$$a+\frac{19}{10}b-bc\leq 2-\sqrt 2$$$$a+\frac{17}{10}b+c-bc\leq 3$$$$ a^2+\frac{9}{5}b-bc\leq 6-4\sqrt 2$$$$ a^2+\frac{8}{5}b^2-bc\leq 6-4\sqrt 2$$$$a+1.974873b-bc\leq 2-\sqrt 2$$$$a+1.775917b+c-bc\leq 3$$

0 replies
sqing
2 hours ago
0 replies
J
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Two permutations
Nima Ahmadi Pour   12
N 3 hours ago by Zhaom
Source: Iran prepration exam
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 + a_2 + \ldots + a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i - b_i - c_i \]

Proposed by Ricky Liu & Zuming Feng, USA
12 replies
Nima Ahmadi Pour
Apr 24, 2006
Zhaom
3 hours ago
J
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Easy Number Theory
math_comb01   37
N 3 hours ago by John_Mgr
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
37 replies
math_comb01
Jan 21, 2024
John_Mgr
3 hours ago
J
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ALGEBRA INEQUALITY
Tony_stark0094   3
N 3 hours ago by sqing
$a,b,c > 0$ Prove that $$\frac{a^2+bc}{b+c} + \frac{b^2+ac}{a+c} + \frac {c^2 + ab}{a+b} \geq a+b+c$$
3 replies
Tony_stark0094
Today at 12:17 AM
sqing
3 hours ago
J
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Inspired by hlminh
sqing   3
N 3 hours ago by sqing
Source: Own
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that $$ |a-kb|+|b-kc|+|c-ka|\leq \sqrt{3k^2+2k+3}$$Where $ k\geq 0 . $
3 replies
sqing
Yesterday at 4:43 AM
sqing
3 hours ago
J
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A Familiar Point
v4913   51
N 3 hours ago by xeroxia
Source: EGMO 2023/6
Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.
51 replies
v4913
Apr 16, 2023
xeroxia
3 hours ago
J
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Apple sharing in Iran
mojyla222   3
N 3 hours ago by math-helli
Source: Iran 2025 second round p6
Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.
3 replies
mojyla222
Apr 20, 2025
math-helli
3 hours ago
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J
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Two numbers divisible by 2025
nAalniaOMliO   4
N Mar 29, 2025 by nAalniaOMliO
Source: Belarusian National Olympiad 2025
Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$.
4 replies
nAalniaOMliO
Mar 28, 2025
nAalniaOMliO
Mar 29, 2025
J
Two numbers divisible by 2025
G H J
Reveal topic tags
number theory
L
G H BBookmark kLocked kLocked NReply
Source: Belarusian National Olympiad 2025
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nAalniaOMliO
295 posts
nAalniaOMliO
#1 h  Mar 28, 2025, 8:35 PM
Y by
Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$.
This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 10:28 AM
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RagvaloD
4909 posts
RagvaloD
#2 h  Mar 28, 2025, 10:56 PM
Y by
Is it correct condition?
Because it is possible to choose $n=2025a$ for any $a$
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CHESSR1DER
54 posts
CHESSR1DER
#3 h  Mar 29, 2025, 8:02 AM
Y by
It was smallest possible integer.
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RagvaloD
4909 posts
RagvaloD
#4 h  Mar 29, 2025, 9:58 AM
Y by
$2025|(n-1)n(n+1)$ and $2025|n(n+1)(n+2)$ so $2025| 3n(n+1)$ or $675|n(n+1)$

$675=3^3*5^2$ and $\gcd(n,n+1)=1$ so
Case 1: $27|n, 25|n+1 \to 27*25|2n+27 \to n =\frac{27*25(2a-1)-27}{2} = 27 ( 25a-13)$
and minimal such $n$ is $324$
$n(n+1)=324*325=2025*52$ so $2025|n^3-n$, and $2025| (n+1)^3-(n+1)$

Case 2: $27|n+1,25|n \to 27*25|2n-25 \to n = \frac{27*25(2a-1)+25}{2}=25(27a-13)$ and minimal such $n$ is $350$
Case 3:$675|n \to n \geq 675$
Case 4$ 675|n+1 \to n \geq 674$

So minimal $n$ is $324$
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nAalniaOMliO
295 posts
nAalniaOMliO
#5 h  Mar 29, 2025, 10:28 AM
Y by
CHESSR1DER wrote:
It was smallest possible integer.
Yep, thanks!
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