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mods with a twist
sketchydealer05   9
N 25 minutes ago by lakshya2009
Source: EGMO 2023/5
We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its twist $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$.
Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.
9 replies
sketchydealer05
Apr 16, 2023
lakshya2009
25 minutes ago
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[MAIN ROUND STARTS MAY 17] OMMC Year 5
DottedCaculator   54
N Today at 12:14 AM by fuzimiao2013
Hello to all creative problem solvers,

Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists? $\phantom{You lost the game.}$
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?

Check out the fifth annual iteration of the

Online Monmouth Math Competition!

Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.

Our website: https://www.ommcofficial.org/
Our Discord (6000+ members): https://tinyurl.com/joinommc
Test portal: https://ommc-test-portal.vercel.app/

This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.

How hard is it?

We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!

How are the problems?

You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc

How will the test be held?/How do I sign up?

Solo teams?

Test Policy

Timeline:
Main Round: May 17th - May 24th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.

Final Round: May 26th - May 28th
The top placing teams will qualify for this invitational round (5-10 questions). The final round consists of 5-10 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.

Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.

Scoring:

Prizes:

I have more questions. Whom do I ask?

We hope for your participation, and good luck!

OMMC staff

OMMC’S 2025 EVENTS ARE SPONSORED BY:

[list]
[*]Nontrivial Fellowship
[*]Citadel
[*]SPARC
[*]Jane Street
[*]And counting!
[/list]

54 replies
DottedCaculator
Apr 26, 2025
fuzimiao2013
Today at 12:14 AM
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9 JMO<200?
DreamineYT   6
N Yesterday at 5:29 PM by lovematch13
Just wanted to ask
6 replies
DreamineYT
May 10, 2025
lovematch13
Yesterday at 5:29 PM
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camp/class recommendations for incoming freshman
walterboro   8
N Tuesday 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
Tuesday at 10:45 PM
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Cyclic Quad
worthawholebean   130
N Tuesday 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
Tuesday at 9:53 PM
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Circle in a Parallelogram
djmathman   55
N Tuesday 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
Tuesday at 5:47 PM
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[Signups Now!] - Inaugural Academy Math Tournament
elements2015   1
N Tuesday 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
May 12, 2025
Ruegerbyrd
Tuesday at 5:16 PM
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Circle Incident
MSTang   39
N Tuesday 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
Tuesday at 4:56 PM
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Lots of Cyclic Quads
Vfire   104
N May 13, 2025 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
May 13, 2025
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Evan's mean blackboard game
hwl0304   72
N May 13, 2025 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
May 13, 2025
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Points Collinear iff Sum is Constant
djmathman   69
N May 13, 2025 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
May 13, 2025
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no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare   2
N Apr 3, 2025 by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
2 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
Apr 3, 2025
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no numbers of the form 80...01 are squares
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Source: Moldova JTST 2024 P5
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Marius_Avion_De_Vanatoare
55 posts
Marius_Avion_De_Vanatoare
#1 h  Jun 10, 2024, 2:37 PM
Y by
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
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navier3072
119 posts
navier3072
#2 h  Jun 10, 2024, 3:00 PM
Y by
Suppose not. Then, for $k\geq 2$, $80\dots01=8 \cdot 10^k +1=X^2$. Then, \[2^{k+3} \cdot 5^k =(X+1)(X-1) \]Hence,
Case 1: $2^{k+2} \cdot 5^m - 2 \cdot 5^n =2 $
Then, $2^{k+1} 5^m - 5^n =1 $. Since, $n>m$ (by size arguments), $5^m \equiv 1 \pmod 5 \implies m=0$, $n=k$. Thus, $2^{k+1} - 5^{k} =1 $, contradiction by, say, Mihailescu.

Case 2: $2 \cdot 5^m - 2^{k+2} \cdot 5^n =2 $
Then, using similar arguments, since $m>n$, $n=0$ and $m=k$, so $5^{k} - 2^{k+1} =1 $, contradiction.
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EVKV
71 posts
EVKV
#3 h  Apr 3, 2025, 2:21 AM
Y by
FTSOC assume it is possible for some k
$2^{k+3}5^{k}$ +$1$ = $X^{2}$
As $gcd(X+1,X-1) = 2$ Only 2 cases possible
Case 1
$2^{k+2}5^{k} -2 = 2$
which is nonsense modulo 5
Case 2
$25^{k} - 2^{k+2}= 2$
again nonsense for $k \geq 2$ (By Catalans or if u want induction)

For $k = 1 $
801 is not a square
This post has been edited 1 time. Last edited by EVKV, Apr 3, 2025, 2:23 AM
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