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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Summer internships/research opportunists in STEM
o99999   5
N 15 minutes ago by SmartGroot
Hi, I am a current high school student and was looking for internships and research opportunities in STEM. Do you guys know any summer programs that do research such as RSI, but for high school freshmen that are open?
Thanks.
5 replies
+1 w
o99999
Apr 22, 2020
SmartGroot
15 minutes ago
HCSSiM results
SurvivingInEnglish   65
N 37 minutes ago by NoSignOfTheta
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
65 replies
SurvivingInEnglish
Apr 5, 2024
NoSignOfTheta
37 minutes ago
camp/class recommendations for incoming freshman
walterboro   3
N 42 minutes ago by Panda729
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
3 replies
walterboro
4 hours ago
Panda729
42 minutes ago
Jane street swag package? USA(J)MO
arfekete   18
N 4 hours ago by Pengu14
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.
18 replies
arfekete
May 7, 2025
Pengu14
4 hours ago
Another integral limit
RobertRogo   2
N Today at 4:02 PM by Gauler
Source: "Traian Lalescu" student contest 2025, Section A, Problem 3
Let $f \colon [0, \infty) \to \mathbb{R}$ be a function differentiable at 0 with $f(0) = 0$. Find
$$\lim_{n \to \infty} \frac{1}{n} \int_{2^n}^{2^{n+1}} f\left(\frac{\ln x}{x}\right) dx$$
2 replies
1 viewing
RobertRogo
Yesterday at 2:28 PM
Gauler
Today at 4:02 PM
Numerical methods problems
jjfgtuuu   0
Today at 3:18 PM
Given that $x_1 = \dfrac{1}{\sqrt{2}}$, $x_2 = \dfrac{1}{\sqrt{6}}$, $x_3 = \dfrac{1}{\sqrt{8}}$, $x_4 = \dfrac{1}{\sqrt{10}}$.
Find the approximate value of $\mathrm{A} = \sum\limits_{i=1}^{4}x_i $ and its absolute and relative error, known that its absolute error is equal or lower than $10^{-5}.$
0 replies
jjfgtuuu
Today at 3:18 PM
0 replies
Group Theory
Stephen123980   2
N Today at 2:23 PM by BadAtMath23
Let G be a group of order $45.$ If G has a normal subgroup of order $9,$ then prove that $G$ is abelian without using Sylow Theorems.
2 replies
Stephen123980
Yesterday at 5:32 PM
BadAtMath23
Today at 2:23 PM
Double integrals
fermion13pi   0
Today at 1:58 PM
Source: Apostol, vol 2
Evaluate the double integral by converting to polar coordinates:

\[
\int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx
\]
Change the order of integration and then convert to polar coordinates.

0 replies
fermion13pi
Today at 1:58 PM
0 replies
D1028 : A strange result about linear algebra
Dattier   0
Today at 1:49 PM
Source: les dattes à Dattier
Let $p>3$ a prime number, with $H \subset M_p(\mathbb R), \dim(H)\geq 2$ and $H-\{0\} \subset GL_p(\mathbb R)$, $H$ vector space.

Is it true that $H-\{0\}$ is a group?
0 replies
Dattier
Today at 1:49 PM
0 replies
Preparing for Putnam level entrance examinations
Cats_on_a_computer   4
N Today at 1:16 PM by Cats_on_a_computer
Non American high schooler in the equivalent of grade 12 here. Where I live, two the best undergraduates program in the country accepts students based on a common entrance exam. The first half of the exam is “screening”, with 4 options being presented per question, each of which one has to assign a True or False. This first half is about the difficulty of an average AIME, or JEE Adv paper, and it is a requirement for any candidate to achieve at least 24/40 on this half for the examiners to even consider grading the second part. The second part consists of long form questions, and I have, no joke, seen them literally rip off, verbatim, Putnam A6s. Some of the problems are generally standard textbook problems in certain undergrad courses but obviously that doesn’t translate it to being doable for high school students. I’ve effectively got to prepare for a slightly nerfed Putnam, if you will, and so I’ve been looking for resources (not just problems) for Putnam level questions. Does anyone have any suggestions?
4 replies
Cats_on_a_computer
Yesterday at 8:32 AM
Cats_on_a_computer
Today at 1:16 PM
Marginal Profit
NC4723   1
N Today at 10:09 AM by Juno_34
Please help me solve this
1 reply
NC4723
Dec 11, 2015
Juno_34
Today at 10:09 AM
Romania NMO 2023 Grade 11 P1
DanDumitrescu   15
N Today at 5:46 AM by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[
    \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
    \]
15 replies
DanDumitrescu
Apr 14, 2023
anudeep
Today at 5:46 AM
Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas   11
N Today at 5:27 AM by cappucher
Source: Putnam 1990 A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
11 replies
ahaanomegas
Jul 12, 2013
cappucher
Today at 5:27 AM
Putnam 2000 B4
ahaanomegas   6
N Today at 1:53 AM by mqoi_KOLA
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.
6 replies
ahaanomegas
Sep 6, 2011
mqoi_KOLA
Today at 1:53 AM
9 Practice AIME Exam
Melissa.   22
N Apr 12, 2025 by jb2015007
(This practice test is designed to be slightly harder than the real test. I would recommend you take this like a real test, using a 3 hour time limit and no calculator.)

Let me know any suggestions for improvement on test quality, difficulty, problem selection, problem placement, test topics, etc. for the next tests that I make!

Practice AIME

1.
Positive integers a, b, and c satisfy a + b + c = 49 and ab + bc + ca = 471. Find the value of the product abc.

2.
Find the integer closest to the value of (69^(1/2) + 420^(1/2))^2.

3.
Let G and A be two points that are 243 units apart. Suppose A_1 is at G, and for n > 1, A_n is the point on line GA such that A_nA_(n-1) = 243, and A_n is farther from A than G. Let L be the locus of points T such that GT + A_6T = 2025. Find the maximum possible distance from T to line GA as T varies across L.

4.
Find the value of (69 + 12 * 33^(1/2))^(1/2) + (69 - 12 * 33^(1/2))^(1/2).

5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.

6.
Find the limit of (1x^2 + 345x^6)/(5x^6 + 78x + 90) as x approaches infinity.

7.
Find the slope of the line tangent to the graph of y = 6x^2 + 9x + 420 at the point where y = 615 and x is positive.

8.
Find the smallest positive integer n such that the sum of the positive integer divisors of n is 1344.

9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.

10.
Let n be the smallest positive integer in base 10 such that the base 2 expression of 60n contains an odd number of 1’s. Find the sum of the squares of the digits of n.

11.
Find the sum of the 7 smallest positive integers n such that n is a multiple of 7, and the repeating decimal expansion of 1/n does not have a period of 6.

12.
Let n be an integer from 1 to 999, inclusive. How many different numerators are possible when n/1000 is written as a common fraction in lowest terms?

13.
How many ways are there to divide a pile of 15 indistinguishable bricks?

14.
Let n be the unique 3-digit positive integer such that the value of the product 100n can be expressed in bases b, b + 1, b + 2, and b + 3 using only 0’s and 1’s, for some integer b > 1. Find n.

15.
For positive integers n, let f(n) be the sum of the positive integer divisors of n. Suppose a positive integer k is untouchable if there does not exist a positive integer a such that f(a) = k + a. For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.

Answer key:
WARNING: SPOILERS!!!
22 replies
Melissa.
Apr 8, 2025
jb2015007
Apr 12, 2025
Practice AIME Exam
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Melissa.
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9Poll:
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(This practice test is designed to be slightly harder than the real test. I would recommend you take this like a real test, using a 3 hour time limit and no calculator.)

Let me know any suggestions for improvement on test quality, difficulty, problem selection, problem placement, test topics, etc. for the next tests that I make!

Practice AIME

1.
Positive integers a, b, and c satisfy a + b + c = 49 and ab + bc + ca = 471. Find the value of the product abc.

2.
Find the integer closest to the value of (69^(1/2) + 420^(1/2))^2.

3.
Let G and A be two points that are 243 units apart. Suppose A_1 is at G, and for n > 1, A_n is the point on line GA such that A_nA_(n-1) = 243, and A_n is farther from A than G. Let L be the locus of points T such that GT + A_6T = 2025. Find the maximum possible distance from T to line GA as T varies across L.

4.
Find the value of (69 + 12 * 33^(1/2))^(1/2) + (69 - 12 * 33^(1/2))^(1/2).

5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.

6.
Find the limit of (1x^2 + 345x^6)/(5x^6 + 78x + 90) as x approaches infinity.

7.
Find the slope of the line tangent to the graph of y = 6x^2 + 9x + 420 at the point where y = 615 and x is positive.

8.
Find the smallest positive integer n such that the sum of the positive integer divisors of n is 1344.

9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.

10.
Let n be the smallest positive integer in base 10 such that the base 2 expression of 60n contains an odd number of 1’s. Find the sum of the squares of the digits of n.

11.
Find the sum of the 7 smallest positive integers n such that n is a multiple of 7, and the repeating decimal expansion of 1/n does not have a period of 6.

12.
Let n be an integer from 1 to 999, inclusive. How many different numerators are possible when n/1000 is written as a common fraction in lowest terms?

13.
How many ways are there to divide a pile of 15 indistinguishable bricks?

14.
Let n be the unique 3-digit positive integer such that the value of the product 100n can be expressed in bases b, b + 1, b + 2, and b + 3 using only 0’s and 1’s, for some integer b > 1. Find n.

15.
For positive integers n, let f(n) be the sum of the positive integer divisors of n. Suppose a positive integer k is untouchable if there does not exist a positive integer a such that f(a) = k + a. For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.

Answer key:
WARNING: SPOILERS!!!
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jkim0656
998 posts
#2
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what happened to 12 on answer key?
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Melissa.
6 posts
#3
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jkim0656 wrote:
what happened to 12 on answer key?

Oh… I lost the answer to that one. Ima try and re-solve it to see if I can get it again.
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fake123
93 posts
#4
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move this to the mock contests forum
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Tetra_scheme
102 posts
#5
Y by
guys please make mock contests into pdfs
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neeyakkid23
122 posts
#6
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Please latex and put into pdf
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vincentwant
1388 posts
#7
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All of the second half problems are way easier except p14,15 which is just mem
This post has been edited 4 times. Last edited by vincentwant, Apr 8, 2025, 4:56 PM
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Melissa.
6 posts
#8
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vincentwant wrote:
All of the second half problems are way easier except p14,15 which is just mem

Wait what is wrong with 14 and 15
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Andyluo
962 posts
#9 • 1 Y
Y by MathRook7817
latexed using https://www.text2latex.com/

Practice AIME

1.
Positive integers \( a \), \( b \), and \( c \) satisfy \( a + b + c = 49 \) and \( ab + bc + ca = 471 \). Find the value of the product \( abc \).

2.
Find the integer closest to the value of \( (69^{1/2} + 420^{1/2})^2 \).

3.
Let \( G \) and \( A \) be two points that are 243 units apart. Suppose \( A_1 \) is at \( G \), and for \( n > 1 \), \( A_n \) is the point on line \( GA \) such that \( A_nA_{n-1} = 243 \), and \( A_n \) is farther from \( A \) than \( G \). Let \( L \) be the locus of points \( T \) such that \( GT + A_6T = 2025 \). Find the maximum possible distance from \( T \) to line \( GA \) as \( T \) varies across \( L \).

4.
Find the value of \( (69 + 12 \cdot 33^{1/2})^{1/2} + (69 - 12 \cdot 33^{1/2})^{1/2} \).

5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.

6.
Find the limit of \( \frac{1x^2 + 345x^6}{5x^6 + 78x + 90} \) as \( x \) approaches infinity.

7.
Find the slope of the line tangent to the graph of \( y = 6x^2 + 9x + 420 \) at the point where \( y = 615 \) and \( x \) is positive.

8.
Find the smallest positive integer \( n \) such that the sum of the positive integer divisors of \( n \) is 1344.

9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.

10.
Let \( n \) be the smallest positive integer in base 10 such that the base 2 expression of \( 60n \) contains an odd number of 1’s. Find the sum of the squares of the digits of \( n \).

11.
Find the sum of the 7 smallest positive integers \( n \) such that \( n \) is a multiple of 7, and the repeating decimal expansion of \( 1/n \) does not have a period of 6.

12.
Let \( n \) be an integer from 1 to 999, inclusive. How many different numerators are possible when \( n/1000 \) is written as a common fraction in lowest terms?

13.
How many ways are there to divide a pile of $15$ indistinguishable bricks?

14.
Let \( n \) be the unique 3-digit positive integer such that the value of the product \( 100n \) can be expressed in bases \( b \), \( b + 1 \), \( b + 2 \), and \( b + 3 \) using only 0’s and 1’s, for some integer \( b > 1 \). Find \( n \).

15.
For positive integers \( n \), let \( f(n) \) be the sum of the positive integer divisors of \( n \). Suppose a positive integer \( k \) is untouchable if there does not exist a positive integer \( a \) such that \( f(a) = k + a \). For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.
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c_double_sharp
315 posts
#10
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#6 is just the average precalc problem
#7 is free with power rule
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hashbrown2009
190 posts
#11
Y by
I would argue this is actually easier than the recent AIME exams but I still did bad and got 13
I got #6 wrong because I am dumb
I got #15 wrong, #15 is actually decently hard
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sanaops9
833 posts
#12
Y by
personally there's quite a few problems that are straight applications of formulas or concepts (ex. #6, #7, #13, actually these might be the only ones). some of the problems are like aime difficulty tho
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mathprodigy2011
325 posts
#13
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sanaops9 wrote:
personally there's quite a few problems that are straight applications of formulas or concepts (ex. #6, #7, #13, actually these might be the only ones). some of the problems are like aime difficulty tho

yea p6 is not aime style because they wouldnt directly put limits nor do they put just conceptual questions on the test
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mathprodigy2011
325 posts
#14
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mathprodigy2011 wrote:
sanaops9 wrote:
personally there's quite a few problems that are straight applications of formulas or concepts (ex. #6, #7, #13, actually these might be the only ones). some of the problems are like aime difficulty tho

yea p6 is not aime style because they wouldnt directly put limits nor do they put just conceptual questions on the test. Also p7 is just derivatives if u know it.
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hashbrown2009
190 posts
#15
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mathprodigy2011 wrote:
sanaops9 wrote:
personally there's quite a few problems that are straight applications of formulas or concepts (ex. #6, #7, #13, actually these might be the only ones). some of the problems are like aime difficulty tho

yea p6 is not aime style because they wouldnt directly put limits nor do they put just conceptual questions on the test

Agreed.
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kamuii
231 posts
#16
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7 (with a knowledge of derivatives) is rlly high up imo
4 is trivial IA
This post has been edited 1 time. Last edited by kamuii, Apr 9, 2025, 9:42 PM
Reason: f
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mathprodigy2011
325 posts
#17
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and p9 is just outright annoying, not aime style.(also p13 should specify whether just 1 pile counts as dividing) Some feedback for this question set, would be to make very long annoying problems instead of problems that are decently tricky but conceptually easy to understand(lots of aime problems are hard just because it is hard to comprehend)
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fake123
93 posts
#18
Y by
also why si the subject distribution so bad
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NamelyOrange
508 posts
#19
Y by
mathprodigy2011 wrote:
and p9 is just outright annoying, not aime style.(also p13 should specify whether just 1 pile counts as dividing) Some feedback for this question set, would be to make very long annoying problems instead of problems that are decently tricky but conceptually easy to understand(lots of aime problems are hard just because it is hard to comprehend)

I suppose the point of this is using the fractional binomial theorem? It's still pretty annoying with it though...
This post has been edited 1 time. Last edited by NamelyOrange, Apr 10, 2025, 12:07 PM
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mathprodigy2011
325 posts
#20
Y by
NamelyOrange wrote:
mathprodigy2011 wrote:
and p9 is just outright annoying, not aime style.(also p13 should specify whether just 1 pile counts as dividing) Some feedback for this question set, would be to make very long annoying problems instead of problems that are decently tricky but conceptually easy to understand(lots of aime problems are hard just because it is hard to comprehend)

I suppose the point of this is using the fractional binomial theorem? It's still pretty annoying with it though...

yeah its just a lot of work that most people know how to do.
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martianrunner
201 posts
#21
Y by
no regular geo (just two analytical geo problems), which is odd

no complex problems

and no trig problems

wayyyy too much nt

this isnt akin to an aime
This post has been edited 1 time. Last edited by martianrunner, Apr 11, 2025, 6:03 PM
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RandomMathGuy500
58 posts
#22
Y by
way too straightforward math. It's like a school test which it shouldn't be. Especially P8+9
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jb2015007
1948 posts
#23
Y by
bro what is p13 :skull:
so unoriginal lol
even i solved
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