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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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
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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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3^n + 61 is a square
VideoCake   24
N 28 minutes ago by maromex
Source: 2025 German MO, Round 4, Grade 11/12, P6
Determine all positive integers \(n\) such that \(3^n + 61\) is the square of an integer.
24 replies
+1 w
VideoCake
Yesterday at 5:14 PM
maromex
28 minutes ago
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A sharp one with 3 var (3)
mihaig   1
N 30 minutes ago by aaravdodhia
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
1 reply
mihaig
2 hours ago
aaravdodhia
30 minutes ago
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Lines AD, BE, and CF are concurrent
orl   49
N 33 minutes ago by Ilikeminecraft
Source: IMO Shortlist 2000, G3
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.
49 replies
orl
Aug 10, 2008
Ilikeminecraft
33 minutes ago
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Is this F.E.?
Jackson0423   1
N an hour ago by jasperE3

Let the set \( A = \left\{ \frac{f(x)}{x} \;\middle|\; x \neq 0,\ x \in \mathbb{R} \right\} \) be finite.
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) satisfying the following condition for all real numbers \( x \):
\[ f(x - 1 - f(x)) = f(x) - x - 1. \]
1 reply
+1 w
Jackson0423
4 hours ago
jasperE3
an hour ago
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IMO Shortlist 2014 N2
hajimbrak   31
N an hour ago by Sakura-junlin
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
Proposed by Titu Andreescu, USA
31 replies
hajimbrak
Jul 11, 2015
Sakura-junlin
an hour ago
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Permutations of Integers from 1 to n
Twoisntawholenumber   76
N 2 hours ago by maromex
Source: 2020 ISL C1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.

Proposed by United Kingdom
76 replies
1 viewing
Twoisntawholenumber
Jul 20, 2021
maromex
2 hours ago
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Another right angled triangle
ariopro1387   5
N 2 hours ago by aaravdodhia
Source: Iran Team selection test 2025 - P7
Let $ABC$ be a right angled triangle with $\angle A=90$.Let $M$ be the midpoint of $BC$, and $P$ be an arbitrary point on $AM$. The reflection of $BP$ over $AB$ intersects lines $AC$ and $AM$ at $T$ and $Q$, respectively. The circumcircles of $BPQ$ and $ABC$ intersect again at $F$. Prove that the center of the circumcircle of $CFT$ lies on $BQ$.
5 replies
ariopro1387
May 25, 2025
aaravdodhia
2 hours ago
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trigonometric inequality
MATH1945   8
N 2 hours ago by mihaig
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
8 replies
MATH1945
May 26, 2016
mihaig
2 hours ago
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Inequality
srnjbr   5
N 2 hours ago by mihaig
For real numbers a, b, c and d that a+d=b+c prove the following:
(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)>=0
5 replies
srnjbr
Oct 30, 2024
mihaig
2 hours ago
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IMO 2014 Problem 4
ipaper   170
N 2 hours ago by lpieleanu
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
170 replies
ipaper
Jul 9, 2014
lpieleanu
2 hours ago
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Tilted Students Thoroughly Splash Tiger part 2
DottedCaculator   20
N 2 hours ago by cj13609517288
Source: ELMO 2024/5
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.

Tiger Zhang
20 replies
DottedCaculator
Jun 21, 2024
cj13609517288
2 hours ago
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Prefix sums of divisors are perfect squares
CyclicISLscelesTrapezoid   37
N 2 hours ago by SimplisticFormulas
Source: ISL 2021 N3
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
37 replies
CyclicISLscelesTrapezoid
Jul 12, 2022
SimplisticFormulas
2 hours ago
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Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   3
N 3 hours ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \]Proposed by Pavle Martinović
3 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
3 hours ago
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Equation in integers with gcd and lcm
skellyrah   1
N 3 hours ago by frost23
Find all integers \( x \) and \( y \) such that
\[ \frac{1}{\gcd(x, y)} + \frac{3}{xy} + \frac{y}{\operatorname{lcm}(x, y)} = y, \]where \( \gcd(x, y) \) denotes the greatest common divisor of \( x \) and \( y \), and \( \operatorname{lcm}(x, y) \) denotes their least common multiple.
1 reply
skellyrah
3 hours ago
frost23
3 hours ago
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<ABP=? if <PAC = <PCD = 20^o inside a square ABCD
parmenides51   2
N Aug 21, 2021 by sunken rock
Source: 2014 Cordoba pre-TST for IberoAmerican p4 (Argentina)
Let $ABCD$ be a square. Let $P$ be inside $ABCD$ such that $\angle PAC = \angle PCD = 20^o$. Find $\angle ABP$ .
2 replies
parmenides51
Aug 20, 2021
sunken rock
Aug 21, 2021
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<ABP=? if <PAC = <PCD = 20^o inside a square ABCD
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Reveal topic tags
geometry
angles
square
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Source: 2014 Cordoba pre-TST for IberoAmerican p4 (Argentina)
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parmenides51
30653 posts
parmenides51
#1 h  Aug 20, 2021, 12:48 PM • 1 Y
Y by Niruuuuut2
Let $ABCD$ be a square. Let $P$ be inside $ABCD$ such that $\angle PAC = \angle PCD = 20^o$. Find $\angle ABP$ .
This post has been edited 1 time. Last edited by parmenides51, Aug 20, 2021, 12:48 PM
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AlanLG
241 posts
AlanLG
#2 h  Aug 20, 2021, 4:00 PM
Y by
Let $x=\angle ABP$, it´s easy to obtain $\angle PAB=65^\circ, \angle BPA=115-x, \angle BCP=70^\circ, \angle CPB=20+x$

By law of sines in

$\triangle ABP: \frac{\sin 65^\circ}{BP}=\frac{\sin (115-x)}{AB}$

$\triangle BCP: \frac{BP}{\sin 70^\circ}=\frac{BC}{\sin (20+x)}$

Since $AB=BC$, multiplying the equations we get

$\frac{\sin 65^\circ}{\sin 70^\circ}=\frac{\sin (115-x)}{\sin (20+x)}$

$x=50^\circ$ clearly satifies, thus $x=\angle ABP=50^\circ$
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sunken rock
4401 posts
sunken rock
#3 h  Aug 21, 2021, 11:29 AM
Y by
One liner solution at https://stanfulger.blogspot.com/2021/08/aops-httpsartofproblemsolvingcomcommuni_21.html

Best regards,
sunken rock
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