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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
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[*] 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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Perspective triangles
ferma2000   1
N Sep 21, 2015 by TelvCohl
Dear mathlinkers;
1) $ABC$ is a triangle.
2) The incircle $(I)$ of $\triangle ABC$ meet $BC,CA,AB$ at $D,E,F$ respectively.
3) $\varrho$ denotes a circle passing throw $B,C$ that is tangent to $(I)$.
4) $\varrho\cap AC=S,\varrho\cap AB=T$
5) $J$ denotes the $A$-excenter of $\triangle AST$.
6) $SJ\cap EF=K,TJ\cap EF=L$.
7) $O$ is the circumcenter of $\triangle JKL$.
8) $M$ is midpoint of $JO$.
Claim:
The triangle formed by reflections of $M$ with respect to $AI,BI,CI$ is perspective with $\triangle DEF$.
Best regards;
1 reply
ferma2000
Sep 21, 2015
TelvCohl
Sep 21, 2015
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Perspective triangles
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ferma2000
56 posts
ferma2000
#1 h  Sep 21, 2015, 7:39 AM • 2 Y
Y by Adventure10, Mango247
Dear mathlinkers;
1) $ABC$ is a triangle.
2) The incircle $(I)$ of $\triangle ABC$ meet $BC,CA,AB$ at $D,E,F$ respectively.
3) $\varrho$ denotes a circle passing throw $B,C$ that is tangent to $(I)$.
4) $\varrho\cap AC=S,\varrho\cap AB=T$
5) $J$ denotes the $A$-excenter of $\triangle AST$.
6) $SJ\cap EF=K,TJ\cap EF=L$.
7) $O$ is the circumcenter of $\triangle JKL$.
8) $M$ is midpoint of $JO$.
Claim:
The triangle formed by reflections of $M$ with respect to $AI,BI,CI$ is perspective with $\triangle DEF$.
Best regards;
Attachments:
Z K Y
The post below has been deleted. Click to close.
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TelvCohl
2312 posts
TelvCohl
#2 h  Sep 21, 2015, 10:04 AM • 5 Y
Y by ferma2000, mineiraojose, enhanced, Adventure10, Mango247
Let $ H $ be the orthocenter of $ \triangle DEF $ and $ R $ $ \equiv $ $ DH $ $ \cap $ $ EF $. From Mannheim theorem we know the incenter $ U, $ $ V $ of $ \triangle BCT, $ $ \triangle BCS $ is the midpoint of $ FD, $ $ DE $, respectively. From Thebault-Sawayama theorem we get $ K, $ $ L $ is the incenter of $ \triangle TSC, $ $ \triangle TSB $, respectively, so $ UVKL $ is a rectangle (well-known).

Let $ \omega $ be the image of $ \odot (I) $ under the Inversion with center $ A $ that fixed $ \varrho $. Clearly, $ \omega $ is tangent to $ \varrho, $ $ CA, $ $ AB, $ and $ \odot (ATS) $. Since $ \omega $ is the A-mixtilinear excircle of $ \triangle AST $, so if $ X $ $ \equiv $ $ \omega $ $ \cap $ $ CA, $ $ Y $ $ \equiv $ $ \omega $ $ \cap $ $ AB $ then by Mannheim theorem we get $ J $ is the midpoint of $ XY $. On the other hand, from Thebault-Sawayama theorem $ \Longrightarrow $ $ U, $ $ V, $ $ X, $ $ Y $ are collinear, so $ J $ lies on $ UV $ $ \Longrightarrow $ $ J $ is the projection of $ I $ on $ UV $, hence notice the reflection $ Z $ of $ R $ in $ J $ lies on $ \odot (I) $ we get $ O $ is the midpoint of $ IR $.

Let $ W $ $ \equiv $ $ OJ $ $ \cap $ $ DH $. From $ OI $ $ = $ $ OR $ $ \Longrightarrow $ $ IJ $ $ = $ $ RW $, so $ HW $ $ / $ $ IJ $ $ = $ $ 3 $ $ / $ $ 1 $ $ = $ $ MW $ $ / $ $ MJ $ $ \Longrightarrow $ $ M $ lies on the Euler line $ IH $ of $ \triangle DEF $, hence from with the Euler's line we get the triangle whose vertex are the reflection of $ M $ in the perpendicular bisector $ AI, $ $ BI, $ $ CI $ of $ EF, $ $ FD, $ $ DE $, respectively is perspective with $ \triangle DEF $.
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