MOP Cutoffs Out?

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766 posts

#1 h Apr 1, 2025, 11:02 PM • 22 Y

Y by Andyluo, zhoujef000, williamxiao, scannose, bjump, aidan0626, ARWonder, Liontiger, Alex-131, jkim0656, Yrock, krithikrokcs, justJen, megarnie, OronSH, arfekete, Toinfinity, vincentwant, mdk2013, cubres, RollingPanda4616, mrtheory

MAA has just emailed a press release announcing the formula they will be using this year to come up with the MOP cutoff that applies to you! Here's the process:

1. Multiply your age by $1434$ , let $n$ be the result.

2. Calculate $\varphi(n)$ , where $\varphi$ is the Euler's totient theorem, which calculates the number of integers less than $n$ relatively prime to $n$ .

3. Multiply your result by $1434$ again because why not, let the result be $m$ .

4. Define the Fibonacci sequence $F_0 = 1, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ for $n \ge 2$ . Let $r$ be the remainder $F_m$ leaves when you divide it by $69$ .

5. Let $x$ be your predicted USA(J)MO score.

6. You will be invited if your score is at least $\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$ .

7. Note that there may be additional age restrictions for non-high schoolers.

See here for MAA's original news message.

.

.

.

Edit (4/2/2025): This was an April Fool's post.
Here's the punchline

This post has been edited 4 times. Last edited by Mathandski, Apr 2, 2025, 4:38 PM

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sadas123

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sadas123

#2 h Apr 1, 2025, 11:24 PM • 1 Y

Y by cubres

Imagine knowing that the video was a Rick roll and still pressing it :skull

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Pengu14

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Pengu14

#3 h Apr 1, 2025, 11:26 PM • 1 Y

Y by cubres

I got r=1. Does this mean I’ve made it!?

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MathRook7817

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MathRook7817

#4 h Apr 1, 2025, 11:28 PM • 2 Y

Y by Pengu14, cubres

haha u cant fool me
i got the url basically memorized

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williamxiao

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williamxiao

#5 h Apr 1, 2025, 11:47 PM • 1 Y

Y by cubres

MathRook7817 wrote:

haha u cant fool me
i got the url basically memorized

whoa why does the color change even in the screenshot

Is it even possible for $x < \lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$
If we ignore the floor, for all r>1, $\sqrt[r]{r^2} \le r$ so $\frac{\sqrt[r]{r^2}}{r}$ contributes <1, $\frac{x}{r} < x-1$ , and $\frac{r \ln(r)}{r} = \ln(r)$ grows very slowly compared to $x-\frac{x}{r}$

Conclusion: We all made MOP!!!!

This post has been edited 1 time. Last edited by williamxiao, Apr 1, 2025, 11:48 PM

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Pengu14

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Pengu14

#6 h Apr 1, 2025, 11:52 PM • 1 Y

Y by cubres

It’s possible for our score to be significantly lower than prediction though

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williamxiao

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williamxiao

#7 h Apr 1, 2025, 11:54 PM • 1 Y

Y by cubres

Pengu14 wrote:

It’s possible for our score to be significantly lower than prediction though

oh that's true

Well i have a predicted score of 0 because i didn't qual so... guaranteed mop!!!

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Pengu14

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Pengu14

#8 h Apr 1, 2025, 11:55 PM • 1 Y

Y by cubres

williamxiao wrote:

Pengu14 wrote:

It’s possible for our score to be significantly lower than prediction though

oh that's true

Well i have a predicted score of 0 because i didn't qual so... guaranteed mop!!!

No dividing by zero!!!!!

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Mathandski

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Mathandski

#9 h Apr 2, 2025, 12:15 AM • 1 Y

Y by cubres

williamxiao wrote:

MathRook7817 wrote:

haha u cant fool me
i got the url basically memorized

whoa why does the color change even in the screenshot

Is it even possible for $x < \lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$
If we ignore the floor, for all r>1, $\sqrt[r]{r^2} \le r$ so $\frac{\sqrt[r]{r^2}}{r}$ contributes <1, $\frac{x}{r} < x-1$ , and $\frac{r \ln(r)}{r} = \ln(r)$ grows very slowly compared to $x-\frac{x}{r}$

Conclusion: We all made MOP!!!!

As a hint, $r = 1$ happens quite a lot (but not always). As a math problem, feel free to try figuring out the possible ages where $r \neq 1$

This post has been edited 1 time. Last edited by Mathandski, Apr 2, 2025, 12:18 AM

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MathRook7817

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#10 h Apr 2, 2025, 12:21 AM • 1 Y

Y by cubres

lets go i made mop!

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sadas123

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sadas123

#11 h Apr 2, 2025, 12:28 AM • 1 Y

Y by cubres

MathRook7817 wrote:

lets go i made mop!

Me too! Except I didn't make USAJMO

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jkim0656

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jkim0656

#12 h Apr 2, 2025, 1:30 AM • 1 Y

Y by cubres

i made MOP!
woo hoo!

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Mathandski

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Mathandski

#13 h Apr 2, 2025, 1:37 AM • 1 Y

Y by cubres

rbo how are so many people making MOP did I set it up wrong wth

@below I'll let you know tmrw

This post has been edited 1 time. Last edited by Mathandski, Apr 2, 2025, 1:42 AM

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scannose

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scannose

#14 h Apr 2, 2025, 1:41 AM • 1 Y

Y by cubres

Mathandski wrote:

rbo how are so many people making MOP did I set it up wrong wth

is r supposed to be equal to 1

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Richard-Stillhard

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Richard-Stillhard

#15 h Apr 2, 2025, 1:48 AM • 1 Y

Y by cubres

Hello, in step 2 it asks for us to find phi(n), but there are infinitely many integers less than n relatively prime to it... so isn't it infinite? I'm confused.

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lord_of_the_rook

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lord_of_the_rook

#16 h Apr 2, 2025, 1:50 AM • 1 Y

Y by cubres

Richard-Stillhard wrote:

Hello, in step 2 it asks for us to find phi(n), but there are infinitely many integers less than n relatively prime to it... so isn't it infinite? I'm confused.

I think they made a mistake, it means positive integers.

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Richard-Stillhard

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Richard-Stillhard

#17 h Apr 2, 2025, 2:03 AM • 2 Y

Y by lord_of_the_rook, cubres

Yeah I gathered lol, I was being sarcastic.

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Yrock

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Yrock

#18 h Apr 2, 2025, 2:07 AM • 1 Y

Y by cubres

I need help calculating, what's the cycle of the fibonacci numbers modulo 69?

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Yrock

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Yrock

#19 h Apr 2, 2025, 2:15 AM • 1 Y

Y by cubres

I hope I know how to add :|

Anyone verify? Hope I can calculate my score soon!

(also I lost the game)

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Amkan2022

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Amkan2022

#20 h Apr 2, 2025, 2:20 AM • 1 Y

Y by cubres

Looks like my cutoff is... Undefined?

Click to reveal hidden text

This post has been edited 1 time. Last edited by Amkan2022, Apr 2, 2025, 2:21 AM

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Yrock

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Yrock

#21 h Apr 2, 2025, 2:21 AM • 1 Y

Y by cubres

Same here

my r is 0 and my x is 0 :stretcher:

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Yrock

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#22 h Apr 2, 2025, 2:34 AM • 2 Y

Y by Mathandski, cubres

Hope this is the right code! If you are lazy run this..

import math
def totient(n):
    if n == 1:
        return 1
 
    phi = n
    p = 2
    while p * p <= n:
        if n % p == 0:
            while n % p == 0:
                n //= p
            phi -= phi // p
        p += 1
 
    if n > 1:
        phi -= phi // n
 
    return phi
 
fibMod69 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 20, 6, 26, 32, 58, 21, 10, 31, 41, 3, 44, 47, 22, 0, 22, 22, 44, 66, 41, 38, 10, 48, 58, 37, 26, 63, 20, 14, 34, 48, 13, 61, 5, 66, 2, 68, 1, 0]
age = int(input("What is your age?"))
jmoscore = int(input("What is your predicted AMO/JMO score?"))
if age > 20 or age < 12:
    print("Sorry, are not eligible for the MOP at this age.")
elif jmoscore > 42 or jmoscore < 0:
    print("That is not a valid JMO score.")
else:
    n = 1434 * age
    p = totient(n)
    m = 1434 * p
    nr = m%48
    r = fibMod69[nr]
    final = math.floor((jmoscore+r**(2/r)+r*math.log(r))/(r))
    print(str(final)+" compared to "+str(jmoscore))
    if final<=jmoscore:
        print("You made MOP!!!!")
    else:
        print("Womp womp, you failed...")

import math
def totient(n):
    if n == 1:
        return 1
    
    phi = n
    p = 2
    while p * p <= n:
        if n % p == 0:
            while n % p == 0:
                n //= p
            phi -= phi // p
        p += 1
    
    if n > 1:
        phi -= phi // n
    
    return phi

fibMod69 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 20, 6, 26, 32, 58, 21, 10, 31, 41, 3, 44, 47, 22, 0, 22, 22, 44, 66, 41, 38, 10, 48, 58, 37, 26, 63, 20, 14, 34, 48, 13, 61, 5, 66, 2, 68, 1, 0]
age = int(input("What is your age?"))
jmoscore = int(input("What is your predicted AMO/JMO score?"))
if age > 20 or age < 12:
    print("Sorry, are not eligible for the MOP at this age.")
elif jmoscore > 42 or jmoscore < 0:
    print("That is not a valid JMO score.")
else:
    n = 1434 * age
    p = totient(n)
    m = 1434 * p
    nr = m%48
    r = fibMod69[nr]
    final = math.floor((jmoscore+r**(2/r)+r*math.log(r))/(r))
    print(str(final)+" compared to "+str(jmoscore))
    if final<=jmoscore:
        print("You made MOP!!!!")
    else:
        print("Womp womp, you failed...")

Is it just me or can no one qual MOP with this

This post has been edited 4 times. Last edited by Yrock, Apr 2, 2025, 3:04 AM

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Mathandski

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Mathandski

#23 h Apr 2, 2025, 2:45 AM • 1 Y

Y by cubres

Most of the code should be correct but I believe line 30 should say
print(final <= jmoscore)

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Yrock

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Yrock

#24 h Apr 2, 2025, 2:57 AM • 1 Y

Y by cubres

Mathandski wrote:

Most of the code should be correct but I believe line 30 should say
print(final <= jmoscore)

oof edited

Time to change to code so that people don't enter incorrect information

EDIT:

is final always one more than jmoscore?!

This post has been edited 2 times. Last edited by Yrock, Apr 2, 2025, 3:05 AM

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smbellanki

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smbellanki

#25 h Apr 2, 2025, 3:29 AM • 1 Y

Y by cubres

Yrock wrote:

Mathandski wrote:

Most of the code should be correct but I believe line 30 should say
print(final <= jmoscore)

oof edited

Time to change to code so that people don't enter incorrect information

EDIT:

is final always one more than jmoscore?!

Yeah

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Mathandski

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#26 h Apr 2, 2025, 4:33 PM • 10 Y

Y by fake123, cj13609517288, bjump, OronSH, Sedro, blueprimes, Yrock, EpicBird08, cubres, Pengu14

In case anyone was curious, here's the joke: according to the rules, everybody missed MOP by 1 point!

We claim that $r = 1$ if and only if your age cannot be written as $3^a 239^b$ . In other words, unless if you are $1, 3, 9, 27, 81, 239, 243, 717, 729, \dots$ years old, $r = 1$ .

The proof comes down to proving two these parts:
Period of F_n modulo 69 is 48

m is a multiple of 48

Unless if you are $1, 3, 9, 27, 81, 239, 243, \dots$ years old, $48 \mid m$ meaning $F_m \equiv F_0 \equiv 1 \pmod{69}$ . With $r = 1$ proved, we plug it in to see the cutoff is $x + 1$ , which is exactly one more than your USA(J)MO score!

This post has been edited 3 times. Last edited by Mathandski, Apr 2, 2025, 5:24 PM
Reason: = -> \equiv

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BS2012

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BS2012

#27 h Apr 2, 2025, 7:35 PM • 1 Y

Y by cubres

Mathandski wrote:

In case anyone was curious, here's the joke: according to the rules, everybody missed MOP by 1 point!

We claim that $r = 1$ if and only if your age cannot be written as $3^a 239^b$ . In other words, unless if you are $1, 3, 9, 27, 81, 239, 243, 717, 729, \dots$ years old, $r = 1$ .

The proof comes down to proving two these parts:
Period of F_n modulo 69 is 48

m is a multiple of 48

Unless if you are $1, 3, 9, 27, 81, 239, 243, \dots$ years old, $48 \mid m$ meaning $F_m \equiv F_0 \equiv 1 \pmod{69}$ . With $r = 1$ proved, we plug it in to see the cutoff is $x + 1$ , which is exactly one more than your USA(J)MO score!

what if you are 9 years old, or predict way low

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anticodon

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anticodon

#28 h Apr 2, 2025, 10:10 PM • 1 Y

Y by cubres

Mathandski wrote:

1. Multiply your age by $1434$ , let $n$ be the result.

that's when I realized April fools
MAA doesn't age discriminate like that (even if they did, why such a big factor of 1434)

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Yrock

1294 posts

Yrock

#29 h Apr 2, 2025, 10:36 PM • 1 Y

Y by cubres

BS2012 wrote:

Mathandski wrote:

In case anyone was curious, here's the joke: according to the rules, everybody missed MOP by 1 point!

We claim that $r = 1$ if and only if your age cannot be written as $3^a 239^b$ . In other words, unless if you are $1, 3, 9, 27, 81, 239, 243, 717, 729, \dots$ years old, $r = 1$ .

The proof comes down to proving two these parts:
Period of F_n modulo 69 is 48

m is a multiple of 48

Unless if you are $1, 3, 9, 27, 81, 239, 243, \dots$ years old, $48 \mid m$ meaning $F_m \equiv F_0 \equiv 1 \pmod{69}$ . With $r = 1$ proved, we plug it in to see the cutoff is $x + 1$ , which is exactly one more than your USA(J)MO score!

what if you are 9 years old, or predict way low

I'm nine so I made MOP