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Proving ZA=ZB
nAalniaOMliO   8
N 41 minutes ago by Mathgloggers
Source: Belarusian National Olympiad 2025
Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$.
Prove that $ZA=ZH$.
Vadzim Kamianetski
8 replies
nAalniaOMliO
Mar 28, 2025
Mathgloggers
41 minutes ago
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Hard geometry
Lukariman   1
N 41 minutes ago by Lukariman
Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.
1 reply
Lukariman
an hour ago
Lukariman
41 minutes ago
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camp/class recommendations for incoming freshman
walterboro   8
N Yesterday at 10:45 PM by lu1376091
hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty
8 replies
walterboro
May 10, 2025
lu1376091
Yesterday at 10:45 PM
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Cyclic Quad
worthawholebean   130
N Yesterday at 9:53 PM by Mathandski
Source: USAMO 2008 Problem 2
Let $ ABC$ be an acute, scalene triangle, and let $ M$, $ N$, and $ P$ be the midpoints of $ \overline{BC}$, $ \overline{CA}$, and $ \overline{AB}$, respectively. Let the perpendicular bisectors of $ \overline{AB}$ and $ \overline{AC}$ intersect ray $ AM$ in points $ D$ and $ E$ respectively, and let lines $ BD$ and $ CE$ intersect in point $ F$, inside of triangle $ ABC$. Prove that points $ A$, $ N$, $ F$, and $ P$ all lie on one circle.
130 replies
worthawholebean
May 1, 2008
Mathandski
Yesterday at 9:53 PM
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Circle in a Parallelogram
djmathman   55
N Yesterday at 5:47 PM by Ilikeminecraft
Source: 2022 AIME I #11
Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

IMAGE
55 replies
djmathman
Feb 9, 2022
Ilikeminecraft
Yesterday at 5:47 PM
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[Signups Now!] - Inaugural Academy Math Tournament
elements2015   1
N Yesterday at 5:16 PM by Ruegerbyrd
Hello!

Pace Academy, from Atlanta, Georgia, is thrilled to host our Inaugural Academy Math Tournament online through Saturday, May 31.

AOPS students are welcome to participate online, as teams or as individuals (results will be reported separately for AOPS and Georgia competitors). The difficulty of the competition ranges from early AMC to mid-late AIME, and is 2 hours long with multiple sections. The format is explained in more detail below. If you just want to sign up, here's the link:

https://forms.gle/ih548axqQ9qLz3pk7

If participating as a team, each competitor must sign up individually and coordinate team names!

Detailed information below:

Divisions & Teams
[list]
[*] Junior Varsity: Students in 10th grade or below who are enrolled in Algebra 2 or below.
[*] Varsity: All other students.
[*] Teams of up to four students compete together in the same division.
[list]
[*] (If you have two JV‑eligible and two Varsity‑eligible students, you may enter either two teams of two or one four‑student team in Varsity.)
[*] You may enter multiple teams from your school in either division.
[*] Teams need not compete at the same time. Each individual will complete the test alone, and team scores will be the sum of individual scores.
[/list]
[/list]
Competition Format
Both sections—Sprint and Challenge—will be administered consecutively in a single, individually completed 120-minute test. Students may allocate time between the sections however they wish to.

[list=1]
[*] Sprint Section
[list]
[*] 25 multiple‑choice questions (five choices each)
[*] recommended 2 minutes per question
[*] 6 points per correct answer; no penalty for guessing
[/list]

[*] Challenge Section
[list]
[*] 18 open‑ended questions
[*] answers are integers between 1 and 10,000
[*] recommended 3 or 4 minutes per question
[*] 8 points each
[/list]
[/list]
You may use blank scratch/graph paper, rulers, compasses, protractors, and erasers. No calculators are allowed on this examination.

Awards & Scoring
[list]
[*] There are no cash prizes.
[*] Team Awards: Based on the sum of individual scores (four‑student teams have the advantage). Top 8 teams in each division will be recognized.
[*] Individual Awards: Top 8 individuals in each division, determined by combined Sprint + Challenge scores, will receive recognition.
[/list]
How to Sign Up
Please have EACH STUDENT INDIVIDUALLY reserve a 120-minute window for your team's online test in THIS GOOGLE FORM:
https://forms.gle/ih548axqQ9qLz3pk7
EACH STUDENT MUST REPLY INDIVIDUALLY TO THE GOOGLE FORM.
You may select any slot from now through May 31, weekdays or weekends. You will receive an email with the questions and a form for answers at the time you receive the competition. There will be a 15-minute grace period for entering answers after the competition.
1 reply
elements2015
Monday at 8:13 PM
Ruegerbyrd
Yesterday at 5:16 PM
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Circle Incident
MSTang   39
N Yesterday at 4:56 PM by Ilikeminecraft
Source: 2016 AIME I #15
Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.
39 replies
MSTang
Mar 4, 2016
Ilikeminecraft
Yesterday at 4:56 PM
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Lots of Cyclic Quads
Vfire   104
N Yesterday at 5:53 AM by Ilikeminecraft
Source: 2018 USAMO #5
In convex cyclic quadrilateral $ABCD$, we know that lines $AC$ and $BD$ intersect at $E$, lines $AB$ and $CD$ intersect at $F$, and lines $BC$ and $DA$ intersect at $G$. Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^\circ$.

Proposed by Kada Williams
104 replies
Vfire
Apr 19, 2018
Ilikeminecraft
Yesterday at 5:53 AM
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Evan's mean blackboard game
hwl0304   72
N Yesterday at 3:26 AM by HamstPan38825
Source: 2019 USAMO Problem 5, 2019 USAJMO Problem 6
Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps.

Proposed by Yannick Yao
72 replies
hwl0304
Apr 18, 2019
HamstPan38825
Yesterday at 3:26 AM
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Points Collinear iff Sum is Constant
djmathman   69
N Yesterday at 1:37 AM by blueprimes
Source: USAMO 2014, Problem 3
Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.
69 replies
djmathman
Apr 29, 2014
blueprimes
Yesterday at 1:37 AM
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Jane street swag package? USA(J)MO
arfekete   30
N Yesterday at 12:32 AM by NoSignOfTheta
Hey! People are starting to get their swag packages from Jane Street for qualifying for USA(J)MO, and after some initial discussion on what we got, people are getting different things. Out of curiosity, I was wondering how they decide who gets what.
Please enter the following info:

- USAMO or USAJMO
- Grade
- Score
- Award/Medal/HM
- MOP (yes or no, if yes then color)
- List of items you got in your package

I will reply with my info as an example.
30 replies
arfekete
May 7, 2025
NoSignOfTheta
Yesterday at 12:32 AM
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ranttttt
alcumusftwgrind   40
N Monday at 8:02 PM by ZMB038
rant
40 replies
alcumusftwgrind
Apr 30, 2025
ZMB038
Monday at 8:02 PM
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J
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Prove that line TA',OY,MX are concurrent
Bunrong123   1
N Dec 30, 2019 by hectorraul
Let the incircle and the $A$-mixtilinear incircle of a triangle ABC touch $AC, AB$ at $E, F$ and $K, J$ resp.$EF$ and $JK$ meet BC at $X, Y$ resp. The $A$-mixtilinear incircle touches the circumcircle of $ABC$ at $T$ and the reflection of $A'$ in $O$, the circumcenter is $A'$.The midpoint of arc $BAC$ is $M$.Prove that the lines $TA', OY, MX$ are concurrent.
1 reply
Bunrong123
Dec 30, 2019
hectorraul
Dec 30, 2019
J
Prove that line TA',OY,MX are concurrent
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Reveal topic tags
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Bunrong123
79 posts
Bunrong123
#1 h  Dec 30, 2019, 5:29 AM • 1 Y
Y by Adventure10
Let the incircle and the $A$-mixtilinear incircle of a triangle ABC touch $AC, AB$ at $E, F$ and $K, J$ resp.$EF$ and $JK$ meet BC at $X, Y$ resp. The $A$-mixtilinear incircle touches the circumcircle of $ABC$ at $T$ and the reflection of $A'$ in $O$, the circumcenter is $A'$.The midpoint of arc $BAC$ is $M$.Prove that the lines $TA', OY, MX$ are concurrent.
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hectorraul
363 posts
hectorraul
#2 h  Dec 30, 2019, 9:28 AM • 3 Y
Y by Bunrong123, Adventure10, Mango247
I will skip many details,

Let $N=A'I\cap (O)$, $D$ where the incircle touches $BC$, $Z=A'T\cap MN$ and $M_A, M_B, M_C$ the midpoints of the arcs $BC, CA, AB$.

1- $NAFIE$ is cyclic and there is a spiral similarity centered at $N$ sending $(E,B)$ to $(F,C)$. Actually, $XNEB$ and $XNFC$ are cyclics and then $M,N,X$ are collinear.

2- Homotecy at $T$ sending $A-mix$ to $(O)$ shows that $T,J, M_B$ and $T,K,M_C$ are collinear.

3- Pascal theorem shows that $K,I,J$ are collinear and then I is the midpoint of $KJ$.

4- In $\triangle TKJ$, $TA$ is symmedian and $TI$ is median, then $\angle JTI=\angle ATK$ with some angle working we can deduce that $T,I,M$ are collinear.

5- Inversion at $M_A$ with radious $M_AB$. $T$ is the intersection of $(O)$ and the circle of diameter $IM_A$, then the image of $T$ is the intersection of $BC$ and $JK$ which is $Y$. Conclusion $M_A, T, Y$ are collinear.

6- With angle working we get $\angle DIM_A=\angle M_AAA'=\angle M_ANA'$, then with the previous inversion $M_A, D, N$ are collinear and also $INYTD$ is cyclic, then $A,N,Y$ are collinear.

7- Finally Pascal on $(O)$. The intersections $Z=MN\cap A'T$, $Y= AN\cap M_AT$ and $O=AA'\cap MM_A$ are collinear and we are done.
This post has been edited 2 times. Last edited by hectorraul, Dec 30, 2019, 9:33 AM
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