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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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About my new website
Samujjal101   18
N 7 minutes ago by vadava_lx
Hi everybody!
I'm registering some of the finest minds in math into my website.. it's not completely developed.. but still if you want we would be very grateful to have you!
Text to display
Maths-matchmaker is a website for connecting math minds together with a mission to unite together. It uses a matching algorithm to match 1:1 with like minded peers based on their interests or topics in math
18 replies
Samujjal101
Yesterday at 2:28 PM
vadava_lx
7 minutes ago
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Geometry
AlexCenteno2007   1
N an hour ago by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
1 reply
AlexCenteno2007
Yesterday at 3:59 PM
AlexCenteno2007
an hour ago
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Product of all even divisors
girishpimoli   4
N an hour ago by williamxiao
$(1)$ Product of all even divisors of $9000$

$(2)$ If $4$ dice are rolled, Then number of ways of getting sum at least $13$ is
4 replies
girishpimoli
Yesterday at 2:13 PM
williamxiao
an hour ago
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Basic geometry
AlexCenteno2007   3
N an hour ago by AlexCenteno2007
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
3 replies
AlexCenteno2007
Feb 9, 2025
AlexCenteno2007
an hour ago
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Symmedian line
April   91
N 3 hours ago by BS2012
Source: All Russian Olympiad - Problem 9.2, 10.2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
91 replies
April
May 10, 2009
BS2012
3 hours ago
J
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The antipolar lines with respect to a fixed point of a pencil of conics
lxhoanghsgs   0
3 hours ago
Source: Well-known online.
The following problem is well-known online, but as far as I am aware of, there is no synthetic proof of this result. Should anybody know about this result, please give me more information on this (e.g., names of the theorems (if any), or proofs). Thank you in advance!

"Suppose that $A_1, A_2, A_3, A_4$ are four given points on the plane, so that no three of them are collinear. Let $S$ be the set of conics passing through $A_1, A_2, A_3, A_4$. Consider a fixed point $P$, for each $\mathcal{C}\in S$, suppose there are distinct points $A_{\mathcal{C}}, B_{\mathcal{C}}, C_{\mathcal{C}}, D_{\mathcal{C}} \in \mathcal{C}$, so that $P\in A_{\mathcal{C}}B_{\mathcal{C}}, P\in C_{\mathcal{C}}D_{\mathcal{C}}$. Let $l_{\mathcal{C}}$ be the line joining the intersection of $A_{\mathcal{C}}C_{\mathcal{C}}$ and $B_{\mathcal{C}}D_{\mathcal{C}}$ with the intersection of $A_{\mathcal{C}}D_{\mathcal{C}}$ and $B_{\mathcal{C}}C_{\mathcal{C}}$.

1. Prove that the definition of $l_{\mathcal{C}}$ does not depend on the choice of $A_{\mathcal{C}}, B_{\mathcal{C}}, C_{\mathcal{C}}, D_{\mathcal{C}} \in \mathcal{C}$.
2. Prove that $l_{\mathcal{C}}$ passes through a fixed point when $\mathcal{C}$ varies."

The "Generalized problem" in #2 of this post is my attempt for synthetically proving this result, using only cross-ratios and Pascal's theorem.

Sincerely,
XH
0 replies
lxhoanghsgs
3 hours ago
0 replies
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USAMO 2000 Problem 5
MithsApprentice   22
N 4 hours ago by Maximilian113
Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$
22 replies
MithsApprentice
Oct 1, 2005
Maximilian113
4 hours ago
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Common external tangents of two circles
a1267ab   55
N 4 hours ago by awesomeming327.
Source: USA Winter TST for IMO 2020, Problem 2, by Merlijn Staps
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.

Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.

Merlijn Staps
55 replies
a1267ab
Dec 16, 2019
awesomeming327.
4 hours ago
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Trillium geometry
Assassino9931   4
N Yesterday at 8:44 PM by Rayvhs
Source: Bulgaria EGMO TST 2018 Day 2 Problem 1
The angle bisectors at $A$ and $C$ in a non-isosceles triangle $ABC$ with incenter $I$ intersect its circumcircle $k$ at $A_0$ and $C_0$, respectively. The line through $I$, parallel to $AC$, intersects $A_0C_0$ at $P$. Prove that $PB$ is tangent to $k$.
4 replies
Assassino9931
Feb 3, 2023
Rayvhs
Yesterday at 8:44 PM
J
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Similarity through arc midpoint in right triangle
cjquines0   11
N Yesterday at 8:38 PM by ItsBesi
Source: Iranian Geometry Olympiad 2016 Medium 4
Let $\omega$ be the circumcircle of right-angled triangle $ABC$ ($\angle A = 90^{\circ}$). The tangent to $\omega$ at point $A$ intersects the line $BC$ at point $P$. Suppose that $M$ is the midpoint of the minor arc $AB$, and $PM$ intersects $\omega$ for the second time in $Q$. The tangent to $\omega$ at point $Q$ intersects $AC$ at $K$. Prove that $\angle PKC = 90^{\circ}$.

Proposed by Davood Vakili
11 replies
cjquines0
May 26, 2017
ItsBesi
Yesterday at 8:38 PM
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Arbitrary point on BC and its relation with orthocenter
falantrng   21
N Yesterday at 8:18 PM by Mapism
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
21 replies
falantrng
Sunday at 11:47 AM
Mapism
Yesterday at 8:18 PM
J
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Fixed point in a small configuration
Assassino9931   3
N Yesterday at 7:49 PM by dno1467
Source: Balkan MO Shortlist 2024 G3
Let $A, B, C, D$ be fixed points on this order on a line. Let $\omega$ be a variable circle through $C$ and $D$ and suppose it meets the perpendicular bisector of $CD$ at the points $X$ and $Y$. Let $Z$ and $T$ be the other points of intersection of $AX$ and $BY$ with $\omega$. Prove that $ZT$ passes through a fixed point independent of $\omega$.
3 replies
Assassino9931
Sunday at 10:23 PM
dno1467
Yesterday at 7:49 PM
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Another two parallels
jayme   2
N Yesterday at 5:36 PM by jayme
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. M the foot of the perpendicular to (AB) through P.
7. Y the point of intersection of the segment MC and (A)
8. X the point of intersection de AY and BC.

Prove : QX is parallel to AB.

Jean-Louis
2 replies
jayme
Yesterday at 9:21 AM
jayme
Yesterday at 5:36 PM
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AT // BC wanted
parmenides51   103
N Yesterday at 4:12 PM by reni_wee
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
103 replies
parmenides51
Sep 22, 2020
reni_wee
Yesterday at 4:12 PM
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IOQM 2021 P18
Dr_Vex   7
N Apr 24, 2021 by flamewavelight
If
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$
7 replies
Dr_Vex
Jan 17, 2021
flamewavelight
Apr 24, 2021
J
IOQM 2021 P18
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Reveal topic tags
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Dr_Vex
562 posts
Dr_Vex
#1 h  Jan 17, 2021, 6:55 AM
Y by
If
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$
This post has been edited 3 times. Last edited by Dr_Vex, Feb 15, 2021, 2:20 PM
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Navansh
980 posts
Navansh
#4 h  Jan 17, 2021, 7:07 AM
Y by
Dr_Vex wrote:
If
$$\Sigma_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b + c$

Question had $a+b$ and answer was $80$
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AKS_9_54_61
332 posts
AKS_9_54_61
#5 h  Jan 17, 2021, 7:09 AM
Y by
$1 + \frac{1}{k^{2}} + \frac{1}{(k+1)^2} = \frac{k^4+2k^3+3k^2+2k+1}{k^2(k+1)^2} =\left( \frac{k^2+k+1}{k(k+1)} \right)^2 $

Hence $$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = \sum_{k=1}^{40}\frac{k^2+k+1}{k(k+1)} = \sum_{k=1}^{40}\left(1+\frac{1}{k}-\frac{1}{k+1}\right) = 40 + \frac{40}{41}$$
Hence $a+b+c = \boxed{121}$
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N1RAV
160 posts
N1RAV
#6 h  Jan 17, 2021, 7:23 AM • 4 Y
Y by angle-bisector, Mango247, Mango247, Mango247
I think we had to find only $a+b$ and not $a+b+c$
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L567
1184 posts
L567
#7 h  Jan 17, 2021, 7:55 AM
Y by
Yeah, so answer is just 80
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Quantum_fluctuations
1282 posts
Quantum_fluctuations
#8 h  Jan 17, 2021, 8:56 AM • 1 Y
Y by Mango247
Dr_Vex wrote:
If
$$\Sigma_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$

This is an old problem from chinese national high school math league.
This post has been edited 1 time. Last edited by Quantum_fluctuations, Jan 17, 2021, 8:57 AM
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pog
4917 posts
pog
#9 h  Mar 14, 2021, 2:54 AM • 3 Y
Y by Mango247, Mango247, Mango247
It's also USAMTS 3/4/11.
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flamewavelight
52 posts
flamewavelight
#10 h  Apr 24, 2021, 6:50 PM
Y by
It was also asked in Common Admission Test (CAT) in 2008.
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