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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
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
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
Circumcircle of XYZ is tangent to circumcircle of ABC
mathuz   39
N a minute ago by zuat.e
Source: ARMO 2013 Grade 11 Day 2 P4
Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.
39 replies
mathuz
May 22, 2013
zuat.e
a minute ago
Arc Midpoints Form Cyclic Quadrilateral
ike.chen   57
N 11 minutes ago by cj13609517288
Source: ISL 2022/G2
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
57 replies
ike.chen
Jul 9, 2023
cj13609517288
11 minutes ago
Complex number
ronitdeb   0
35 minutes ago
Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$. Find all possible values of $z_1^2+z_2^2+...+z_5^2$
0 replies
ronitdeb
35 minutes ago
0 replies
Elementary Problems Compilation
Saucepan_man02   29
N 41 minutes ago by Electrodynamix777
Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2  -ab-bc-cd-d +2/5=0$$
29 replies
Saucepan_man02
May 26, 2025
Electrodynamix777
41 minutes ago
Cup of Combinatorics
M11100111001Y1R   5
N an hour ago by MathematicalArceus
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
5 replies
M11100111001Y1R
May 27, 2025
MathematicalArceus
an hour ago
Generic Real-valued FE
lucas3617   4
N an hour ago by GreekIdiot
$f: \mathbb{R} -> \mathbb{R}$, find all functions where $f(2x+f(2y-x))+f(-x)+f(y)=2f(x)+f(y-2x)+f(2y)$ for all $x$,$y \in \mathbb{R}$
4 replies
lucas3617
Apr 25, 2025
GreekIdiot
an hour ago
Find all possible values of BT/BM
va2010   54
N an hour ago by lpieleanu
Source: 2015 ISL G4
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
54 replies
va2010
Jul 7, 2016
lpieleanu
an hour ago
A Familiar Point
v4913   52
N an hour ago by SimplisticFormulas
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 $\Omega$.
52 replies
v4913
Apr 16, 2023
SimplisticFormulas
an hour ago
Tangential quadrilateral and 8 lengths
popcorn1   72
N an hour ago by cj13609517288
Source: IMO 2021 P4
Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]
Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland
72 replies
popcorn1
Jul 20, 2021
cj13609517288
an hour ago
An algorithm for discovering prime numbers?
Lukaluce   3
N 2 hours ago by TopGbulliedU
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
3 replies
Lukaluce
May 18, 2025
TopGbulliedU
2 hours ago
Random concyclicity in a square config
Maths_VC   5
N 2 hours ago by Royal_mhyasd
Source: Serbia JBMO TST 2025, Problem 1
Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$, and let $N$ be the intersection point of segments $AC$ and $DM$. The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$, $C$, $P$ and $Q$ lie on a single circle.
5 replies
Maths_VC
Tuesday at 7:38 PM
Royal_mhyasd
2 hours ago
Basic ideas in junior diophantine equations
Maths_VC   3
N 2 hours ago by Royal_mhyasd
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
3 replies
Maths_VC
Tuesday at 7:54 PM
Royal_mhyasd
2 hours ago
Prime number theory
giangtruong13   2
N 2 hours ago by RagvaloD
Find all prime numbers $p,q$ such that: $p^2-pq-q^3=1$
2 replies
giangtruong13
3 hours ago
RagvaloD
2 hours ago
Problem 2
delegat   147
N 3 hours ago by math-olympiad-clown
Source: 0
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]

Proposed by Angelo Di Pasquale, Australia
147 replies
delegat
Jul 10, 2012
math-olympiad-clown
3 hours ago
circle geometry showing perpendicularity
Kyj9981   4
N Apr 24, 2025 by cj13609517288
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$, and line $AC$ intersects $\omega_2$ at point $F \neq A$. If $O$ is the circumcenter of $\triangle AEF$, prove that $OB \perp CD$.
4 replies
Kyj9981
Mar 18, 2025
cj13609517288
Apr 24, 2025
circle geometry showing perpendicularity
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Kyj9981
15 posts
#1 • 1 Y
Y by Rounak_iitr
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$, and line $AC$ intersects $\omega_2$ at point $F \neq A$. If $O$ is the circumcenter of $\triangle AEF$, prove that $OB \perp CD$.
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Retemoeg
59 posts
#2 • 1 Y
Y by Kyj9981
Could be resolved nicely through Carnot's theorem.
Denote $M, N$ midpoints of segments $AE, AF$.
We'd have that $ON \perp AF$, $OM \perp AE$.
\[ MA^2 - MD^2 + BD^2 - BC^2 + NC^2 - NA^2 = 0 \]Notice that, by power of a point and Pythagorean's: $MA^2 - MD^2 = (MA - MD)(ME + MD) = -DA\cdot DE = -DB\cdot DC$.
Similarly, $NC^2 - NA^2 = CB\cdot CD$. Thus, the above sum translates to
\[ -DB\cdot DC + CB\cdot CD + BD^2 - BC^2 = (DB - DC)\cdot CD - (DB - DC)\cdot CD = 0 \]And we are done.
This post has been edited 2 times. Last edited by Retemoeg, Mar 18, 2025, 4:55 PM
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Double07
94 posts
#3 • 2 Y
Y by Calamarul, Kyj9981
We will solve this with Moving Points Method:

Fix the $\omega_1$ and $\omega_2$ circles, $O_1$ and $O_2$ their centers and $A$ and $B$ their intersections and we will move point $C$ along $\omega_1$.

First of all, we will prove that $B, O_1, O_2, O$ are concyclic.

Notice that $OO_1$ is the bisector of segment $AE$ and $OO_2$ is the bisector of segment $AF$.

$\widehat{O_1OO_2}=180^\circ-\widehat{EAF}=180^\circ-\widehat{CAD}$.

We want to prove that $\widehat{O_1OO_2}=180^\circ-\widehat{O_1BO_2}\iff \widehat{CAD}=\widehat{O_1BO_2}=\widehat{O_1AO_2}$.

But $\widehat{O_1AO_2}=180^\circ-\widehat{AO_1O_2}-\widehat{AO_2O_1}=180^\circ-\frac{1}{2}\widehat{AO_1B}-\frac{1}{2}\widehat{AO_2B}=180^\circ-\widehat{ACB}-\widehat{ADB}=\widehat{CAD}\quad\blacksquare$

Let now $X$ be the intersection of the perpendicular in $B$ on the $CD$ line with the $(BO_1O_2)$ circle (other than $B$).

We will show there exist projective maps $C\to O$ and $C\to X$ and then we will just have to prove that $O=X$ for $3$ points $C\in\omega_1$.

Since $C\to AC$ is projective ($A$ is fixed), $AC\to O_2O$ is projective ($O_2O$ is the perpendicular from $O_2$ to $AC$) and $O_2O\to O$ is projective (the $(BO_1O_2)$ circle is fixed and $O$ is the intersection of $O_2O$ with it), we have $C\to O$ - projective.

Since $C\to BC$ is projective ($B$ is fixed), $BC\to BX$ is projective ($BX$ is the perpendicular in $B$ on line $BC$) and $BX\to X$ is projective (the $(BO_1O_2)$ circle is fixed and $X$ is the intersection of $BX$ with it), we have $C\to X$ - projective.

Now we are just left to show that $O=X$ for $3$ points $C$.

1. $C\to B\implies BC$ becomes the tangent to $\omega_1$ in $B\implies BX\to BO_1\implies X\to O_1$.
$C\to B\implies AC\to AB\implies O_2O\to O_2O_1\implies O\to O_1\implies X=O$.

2. $C$ is the antipode of $A$ in $\omega_1\implies BC\parallel O_1O_2\implies BX\to BA\implies X=BA\cap (BO_1O_2)$.
But $\widehat{O_1XO_2}=180^\circ-\widehat{O_1BO_2}=180^\circ-\widehat{O_1AO_2}$.
Since $\widehat{O_1XO_2}=180^\circ-\widehat{O_1AO_2}$ and $XA\perp O_1O_2$, we can conclude that $A$ is the orthocenter of $\Delta XO_1O_2\implies O_1A\perp XO_2$.
We also have $AC=AO_1\implies O_2O\perp O_1A\implies X=O$.

3. $C\to A\implies AC$ becomes the tangent in $A$ at $\omega_1\implies OO_2\parallel O_1A$ and we similarly get $OO_1\parallel O_2A\implies O_1AO_2O$ is a parallelogram. Since $A$ is the reflection of $B$ across $O_1O_2$, is well-known that $O$ is the reflection of $B$ across the bisector of segment $O_1O_2$.
$C\to A\implies BC\to AB\implies BX\parallel O_1O_2\implies O_1O_2XB$ is an isoscelles trapezoid, so $X$ is also the reflection of $B$ across the bisector of segment $O_1O_2$, so $X=O$.

So projective functions $C\to O$ and $C\to X$ are equal in $3$ different points $C$, so $O=X$ for all points $C\in\omega_1$.
This means that $OB\perp CD$.
This post has been edited 1 time. Last edited by Double07, Mar 18, 2025, 7:30 PM
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JollyEggsBanana
2 posts
#4 • 1 Y
Y by Kyj9981
Let $O_1$ and $O_2$ be the centers of $\omega_1$ and $\omega_2$ respectively.

The main claim is that $O_1O_2BO$ is cyclic. Note $O_1O \perp AE$ and $O_2O \perp AF$ by radax. This implies $\measuredangle O_1OO_2 = \measuredangle DAC = \measuredangle O_1BO_2$.

From here we can angle chase
\[\measuredangle OBC = \measuredangle  OBO_1 + \measuredangle O_1BC = \measuredangle FAB + 90 - \measuredangle CAB = 90\]
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cj13609517288
1926 posts
#5 • 1 Y
Y by Kyj9981
https://www.geogebra.org/calculator/vz7v5yep

Let $G=CE\cap DF$ which lies on $(AEF)$ by triangle Miquel. Then $\angle CBE=\angle DBF=\angle G$, so $\angle EOF=2\angle G=180^\circ-\angle EBF$, so $EBFO$ cyclic. Now we can access the argument of $OB$, so the problem dies:
\[\angle OBC=\angle OBE+\angle EBC=\angle OFE+\angle EAC=90^\circ.\]
This post has been edited 1 time. Last edited by cj13609517288, Apr 24, 2025, 2:41 PM
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