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ISI UGB 2025 P7
SomeonecoolLovesMaths   8
N 32 minutes ago by Mathworld314
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
8 replies
SomeonecoolLovesMaths
Yesterday at 11:28 AM
Mathworld314
32 minutes ago
9 Will I make JMO?
EaZ_Shadow   20
N 4 hours ago by Craftybutterfly
will I be able to make it... will the cutoffs will be pre-2024
20 replies
EaZ_Shadow
Feb 7, 2025
Craftybutterfly
4 hours ago
Sequence of integers
tenniskidperson3   32
N 5 hours ago by N3bula
Source: 2012 USAMO problem #3
Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.
32 replies
tenniskidperson3
Apr 24, 2012
N3bula
5 hours ago
Isogonal Conjugates: 2011 USAMO #5
tenniskidperson3   77
N Today at 12:35 AM by N3bula
Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\] Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$.
77 replies
tenniskidperson3
Apr 28, 2011
N3bula
Today at 12:35 AM
camp/class recommendations for incoming freshman
walterboro   7
N Today at 12:33 AM by Ruegerbyrd
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
7 replies
walterboro
Saturday at 6:45 PM
Ruegerbyrd
Today at 12:33 AM
Vertices of a pentagon invariant: 2011 USAMO #2
tenniskidperson3   54
N Yesterday at 11:21 PM by N3bula
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
54 replies
tenniskidperson3
Apr 28, 2011
N3bula
Yesterday at 11:21 PM
HCSSiM results
SurvivingInEnglish   69
N Yesterday at 10:23 PM by MathWizardThatCanBeatYou
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
69 replies
SurvivingInEnglish
Apr 5, 2024
MathWizardThatCanBeatYou
Yesterday at 10:23 PM
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls   1
N Yesterday at 10:09 PM by Indy_Integirls
[center]IMAGE

Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST.
Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]

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[center]Important Information[/center]

Eligibility: This competition is open to all woman-identifying and non-binary students in middle and high school. Non-Indiana residents and international students are welcome as well!

Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.

Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.

Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.

Contact & Connect With Us: Follow us on Instagram @indy.integirls, join our Discord, follow us on TikTok @indy.integirls, and email us at indy@integirls.org.

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[center]Help Us Out

Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
1 reply
Indy_Integirls
Yesterday at 2:36 AM
Indy_Integirls
Yesterday at 10:09 PM
Past USAMO Medals
sdpandit   4
N Yesterday at 8:28 PM by sadas123
Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!
4 replies
sdpandit
May 8, 2025
sadas123
Yesterday at 8:28 PM
Degree Six Polynomial's Roots
ksun48   43
N Yesterday at 6:48 PM by Markas
Source: 2014 AIME I Problem 14
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
43 replies
ksun48
Mar 14, 2014
Markas
Yesterday at 6:48 PM
Jane street swag package? USA(J)MO
arfekete   23
N Yesterday at 6:14 PM by Inaaya
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.
23 replies
arfekete
May 7, 2025
Inaaya
Yesterday at 6:14 PM
Prove excircle is tangent to circumcircle
sarjinius   8
N Apr 24, 2025 by Lyzstudent
Source: Philippine Mathematical Olympiad 2025 P4
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
8 replies
sarjinius
Mar 9, 2025
Lyzstudent
Apr 24, 2025
Prove excircle is tangent to circumcircle
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Source: Philippine Mathematical Olympiad 2025 P4
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sarjinius
241 posts
#1 • 4 Y
Y by MathLuis, mpcnotnpc, JollyEggsBanana, Rounak_iitr
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
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ilovemath0402
188 posts
#2
Y by
bump bump this problem is so nice
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sarjinius
241 posts
#3
Y by
ilovemath0402 wrote:
bump bump this problem is so nice

Thanks, I proposed this problem :)
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SimplisticFormulas
116 posts
#4
Y by
what’s the solution? I am completely stuck
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MathLuis
1525 posts
#5 • 4 Y
Y by drago.7437, sarjinius, Mysteriouxxx, radian_51
Well this geo is really amazing I have to say...solved in around 30 mins but I think this could even be around 30-35 MOHS because the way to find things on this problem requires deep intuition.
Let $BI \cap (ABC)=M_B$ and $CI \cap (ABC)=M_C$, also let $N_A$ be midpoint of arc $BAC$ on $(ABC)$, now let reflections of $D$ over $EX, FY, BI, CI, Y \infty_{\perp CI}, X \infty_{\perp BI}$ be $D_B, D_C, L', K', K, L$ respectively now let reflection of $D_B$ over $AX$ be $L_1$ and reflection of $D_C$ over $AY$ be $K_1$.
Using the paralelogram we can easly see from the direction of the reflections that $KK'$ and $LL'$ are diameters on $(Y, YD), (X, XD)$ respectively, now let $I_B, I_C$ be the $B,C$ excenters of $\triangle ABC$ then notice we have $\measuredangle II_CA=\measuredangle CBI=\measuredangle CDY=\measuredangle YK'C$ which implies $I_CAK'Y$ cyclic and similarily $I_BAL'X$ is cyclic however since $\measuredangle CDY=\measuredangle YD_CF$ we also get that $I_CAK'YD_C$ is cyclic and similarily $I_BAL'XD_B$ is cyclic, however it doesn't end here...
Now notice that $YK'=YD_C$ so $Y$ is midpoint of arc $K'D_C$ on $(I_CAK')$ however $D, K'$ are symetric in $CI$ which means both $I_CD, I_CD'$ are reflections of $I_CK'$ over $CI$ and thus $I_C, D, D_C$ are colinear, and similarily $I_B, D, D_B$ are colinear.
Now $\measuredangle CDY=\measuredangle YD_CA=\measuredangle AK_1Y$ which means $CK_1YD$ is cyclic and similarily we have $L_1BXD$ cyclic, but also note that $\measuredangle L_1DL=\measuredangle L_1L'L=\measuredangle AI_BX=\measuredangle ACI=\measuredangle K_1DY$ which means that $L_1, D, K_1$ are colinear.
Now from here notice that $\measuredangle DL_1A=\measuredangle DXI=\measuredangle IYD=\measuredangle AK_1D$ which does in fact show that $\triangle L_1AK_1$ is isosceles and therefore $AK_1=AL_1$, and from reflections this gives $AD_B=AD_C$, but notice from other reflections we have $D_BG=DG=D_CG$ where $EX \cap FY=G$ (clearly then $G$ is A-excenter of $\triangle EAF$), but now also note that we have $\measuredangle AD_BG=\measuredangle GDE=\measuredangle AD_CG$ which means that $AD_BGD_C$ is cyclic but by summing arcs we end up realising $AG$ is diameter and in fact now this means $(D_BDD_C)$ is $\omega$ from the tangencies.
To finish let $J$ be the miquelpoint of $L_1BCK_1$ then $J$ lies on $(ABC)$ but also from Reim's we get $N_A, D, J$ colinear and then Reim's twice gives $M_CX \cap M_BY=J$ and from double Reim's once again we have that $(AL'X) \cap (AK'Y)=J$ and this is excellent news because now we can note that $\measuredangle D_CJD_B=\measuredangle D_CJA+\measuredangle AJD_B=\measuredangle D_CI_CA+\measuredangle AI_BD_B=\measuredangle D_CDD_B$ which shows that $J$ lies on $\omega$ as well, but since $N_A, D, J$ are colinear from the converse of Archiemedes Lemma (or just shooting Lemma/homothety) we have that $\omega, (ABC)$ are tangent at $J$ as desired thus we are done :cool:
This post has been edited 1 time. Last edited by MathLuis, Mar 13, 2025, 8:16 PM
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AndreiVila
210 posts
#6 • 1 Y
Y by Lyzstudent
Notice that $X$ and $Y$ are the incenters of $\triangle ABE$ and $\triangle ACF$. Let $X'$ and $Y'$ be the projections of $X$ and $Y$ onto $BC$. Let $T$ and $S$ be the projections of $X$ onto $AE$ and $AB$ respectively, and let $K$ be the tangency point of $\omega$ with $AE$.

Claim 1. The $A$-excircle of $\triangle AEF$ is tangent to $EF$ in $D$.
Proof: Since $IXDY$ is a parallelogram, by projecting onto $BC$ we get that $BX'+BY'=BT+BD$. This is equivalent to $$BE+AB-AE+2BF+FC+AF-AC=BA+BC-AC+2BD.$$Simplifying yields $AE+ED=AF+FD$, which is equivalent to $D$ being the tangency point of the excircle.

Claim 2. Circle $\omega$ is tangent to $(ABC)$.
Proof: By Casey's Theorem, we need to prove that $$b\cdot BD + c\cdot CD = a\cdot AK.$$But $$AK=AT+TK=AT+X'D=AT-BX'+BD=AS-BS+BD=c-2BS+BD.$$With Thales' Theorem, $\frac{BS}{p-b}=\frac{BX}{BI}=\frac{BD}{a},$ so $BS=\frac{BD(p-b)}{a},$ thus getting $AK=\frac{ac+BD(b-c)}{a},$ and the conclusion follows.
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SimplisticFormulas
116 posts
#7
Y by
I found that $X,Y$ are in centres, $XE$ meets $YF$ in $Z=$$A$- excentre of $AEF$ and that$A$ appears to be Miquel point of $IXYZ$.
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markam
4 posts
#10
Y by
sarjinius, what solution did you have in mind at first, when you proposed this problem?
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Lyzstudent
1 post
#11
Y by
AndreiVila wrote:
Notice that $X$ and $Y$ are the incenters of $\triangle ABE$ and $\triangle ACF$. Let $X'$ and $Y'$ be the projections of $X$ and $Y$ onto $BC$. Let $T$ and $S$ be the projections of $X$ onto $AE$ and $AB$ respectively, and let $K$ be the tangency point of $\omega$ with $AE$.

Claim 1. The $A$-excircle of $\triangle AEF$ is tangent to $EF$ in $D$.
Proof: Since $IXDY$ is a parallelogram, by projecting onto $BC$ we get that $BX'+BY'=BT+BD$. This is equivalent to $$BE+AB-AE+2BF+FC+AF-AC=BA+BC-AC+2BD.$$Simplifying yields $AE+ED=AF+FD$, which is equivalent to $D$ being the tangency point of the excircle.

Claim 2. Circle $\omega$ is tangent to $(ABC)$.
Proof: By Casey's Theorem, we need to prove that $$b\cdot BD + c\cdot CD = a\cdot AK.$$But $$AK=AT+TK=AT+X'D=AT-BX'+BD=AS-BS+BD=c-2BS+BD.$$With Thales' Theorem, $\frac{BS}{p-b}=\frac{BX}{BI}=\frac{BD}{a},$ so $BS=\frac{BD(p-b)}{a},$ thus getting $AK=\frac{ac+BD(b-c)}{a},$ and the conclusion follows.
Excellent!!!Much better than the solution above.
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