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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
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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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PoP+Parallel
Solilin   1
N 16 minutes ago by sansgankrsngupta
Source: Titu Andreescu, Lemmas in Olympiad Geometry
Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
1 reply
Solilin
44 minutes ago
sansgankrsngupta
16 minutes ago
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gcd of coefficients of polynomial
QueenArwen   2
N 18 minutes ago by AshAuktober
Source: 46th International Tournament of Towns, Senior O-Level P5, Spring 2025
Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.
2 replies
QueenArwen
Mar 11, 2025
AshAuktober
18 minutes ago
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Another config geo with concurrent lines
a_507_bc   15
N 24 minutes ago by E50
Source: BMO SL 2023 G5
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
15 replies
a_507_bc
May 3, 2024
E50
24 minutes ago
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Dwarves at the river
Experia   1
N an hour ago by Radin_
Source: Stage PreIMO 2018 - Italy
There are $100$ dwarves, whose weigths are $1,\,2,\dots,\,100\,\text{kg}$, who want to cross a river. They have a small boat which can lift at most $100\,\text{kg}$ each time without sinking. For each journey of the boat a non-empty subset of the dwarves to be taken to the other side is chosen and one of these dwarves is chosen as the $\emph{rower}$ for that journey. Since return journeys are counter-current, no dwarf is able to do the rower for more than one return journey. Is it possible for all the dwarves to reach the other side of the river?
1 reply
Experia
Apr 23, 2022
Radin_
an hour ago
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Regarding Maaths olympiad prepration
omega2007   4
N an hour ago by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
4 replies
omega2007
Yesterday at 3:13 PM
omega2007
an hour ago
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square root problem that involves geometry
kjhgyuio   2
N 2 hours ago by ND_
If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

2 replies
kjhgyuio
3 hours ago
ND_
2 hours ago
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inquequality
ngocthi0101   9
N 3 hours ago by sqing
let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$
9 replies
ngocthi0101
Sep 26, 2014
sqing
3 hours ago
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Assisted perpendicular chasing
sarjinius   5
N 3 hours ago by hukilau17
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
5 replies
sarjinius
Mar 9, 2025
hukilau17
3 hours ago
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Tangent.
steven_zhang123   2
N 4 hours ago by AshAuktober
Source: China TST 2001 Quiz 6 P1
In \( \triangle ABC \) with \( AB > BC \), a tangent to the circumcircle of \( \triangle ABC \) at point \( B \) intersects the extension of \( AC \) at point \( D \). \( E \) is the midpoint of \( BD \), and \( AE \) intersects the circumcircle of \( \triangle ABC \) at \( F \). Prove that \( \angle CBF = \angle BDF \).
2 replies
steven_zhang123
Mar 23, 2025
AshAuktober
4 hours ago
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IMO ShortList 1998, algebra problem 1
orl   37
N 4 hours ago by Marcus_Zhang
Source: IMO ShortList 1998, algebra problem 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that

\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
37 replies
orl
Oct 22, 2004
Marcus_Zhang
4 hours ago
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Integer Coefficient Polynomial with order
MNJ2357   9
N 4 hours ago by v_Enhance
Source: 2019 Korea Winter Program Practice Test 1 Problem 3
Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.
9 replies
MNJ2357
Jan 12, 2019
v_Enhance
4 hours ago
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Inspired by bamboozled
sqing   0
4 hours ago
Source: Own
Let $ a,b,c $ be reals such that $(a^2+1)(b^2+1)(c^2+1) = 27. $Prove that $$1-3\sqrt 3\leq ab + bc + ca\leq 6$$
0 replies
sqing
4 hours ago
0 replies
J
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Range of ab + bc + ca
bamboozled   1
N 4 hours ago by sqing
Let $(a^2+1)(b^2+1)(c^2+1) = 9$, where $a, b, c \in R$, then the number of integers in the range of $ab + bc + ca$ is __
1 reply
bamboozled
5 hours ago
sqing
4 hours ago
J
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Functional Equation
AnhQuang_67   4
N 4 hours ago by AnhQuang_67
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
4 replies
AnhQuang_67
Yesterday at 4:50 PM
AnhQuang_67
4 hours ago
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Midpoints of altitudes
joh   3
N Mar 11, 2024 by HamstPan38825
Source: RMO 2003, Grade 9, Problem 3
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.

Dorin Popovici
3 replies
joh
Oct 23, 2008
HamstPan38825
Mar 11, 2024
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Midpoints of altitudes
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Reveal topic tags
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Source: RMO 2003, Grade 9, Problem 3
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joh
62 posts
joh
#1 h  Oct 23, 2008, 9:22 AM • 2 Y
Y by Adventure10, Mango247
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.

Dorin Popovici
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joh
62 posts
joh
#2 h  Oct 26, 2008, 7:22 AM • 2 Y
Y by Adventure10, Mango247
Hint
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balan razvan
383 posts
balan razvan
#3 h  Oct 26, 2008, 9:01 AM • 3 Y
Y by galav, Adventure10, Mango247
The middles $ H_1,H_2,H_3$ of the altitudes of the triangle are on the sides of the median triangle.It isn't possible that 2midpoints to be the same.The only posibility to be on the same line is that two of them to form a segment composed by the midpoints of two sides and the third one to be in the interior of this segment.This can hapen only if the triangle has a $ 90$ degree angle.
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HamstPan38825
8857 posts
HamstPan38825
#4 h  Mar 11, 2024, 4:31 PM
Y by
Here's the complex numbers solution from the OTIS walkthrough.

Let $(ABC)$ be the unit circle, as always, and note that the midpoints of the altitudes are $x = \frac 12\left(a+\frac 12\left(a+b+c-\frac{bc}a\right)\right)$ and cyclic permutations. So it suffices to show that $$0 = \begin{vmatrix} 3a+b+c-\frac{bc}a & \frac 3a+\frac 1b+\frac 1c - \frac a{bc} & 1 \\ 3b+a+c-\frac{ac}b & \frac 3b+\frac 1a+\frac 1c-\frac b{ac} & 1 \\ 3c+a+b-\frac{ab}c & \frac 3c+\frac 1a+\frac 1b-\frac c{ab} & 1 \end{vmatrix} = \begin{vmatrix} 2a-\frac{bc}a & \frac 2a - \frac a{bc} & 1 \\ 2b-\frac{ac}b & \frac 2b - \frac b{ac} & 1 \\ 2c-\frac{ab}c & \frac 2c - \frac c{ab} & 1 \end{vmatrix}$$if and only if $0 \in \{a+b, b+c, c+a\}$. It turns out that this factors as $$\sum_{\mathrm{cyc}} \left(2a-\frac{bc}a\right)\left(\frac 2b-\frac b{ac}\right) - \left(\frac 2a-\frac a{bc}\right)\left(2b-\frac{ac}b\right) = -\frac{(a-b)(b-c)(c-a)(a+b)(b+c)(c+a)}{a^2b^2c^2}$$as needed.
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