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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
9 Was this year's AMC 8 hard or easy
ChuMath   65
N 14 minutes ago by giratina3
Question: Was this year's AMC 8 harder or easier than last year's (or the historical average)

65 replies
ChuMath
Jan 25, 2024
giratina3
14 minutes ago
Problem3
samithayohan   120
N 43 minutes ago by LHE96
Source: IMO 2015 problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine
120 replies
samithayohan
Jul 10, 2015
LHE96
43 minutes ago
BMO 2024 SL A1
MuradSafarli   11
N an hour ago by DensSv
Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]
holds for all positive real numbers \( a, b \) and \( c \).
11 replies
1 viewing
MuradSafarli
Apr 27, 2025
DensSv
an hour ago
2500 VIMC mathematics competition
OWOW   45
N an hour ago by OWOW
Hello! This is my second math competition I made up, this time, set in the year 2500 on a Venus base in the upper atmosphere.
This is basically AMC8 for Venusians.

2500 VIMC

What is the sum of all perfect squares less than 2500 in the form $9p^2$ where $p$ is prime?

Equilateral triangle $ABC$ has Area of $6$. If point $D$ is the midpoint of $AB$, then line $L$ is $CD$, which cuts the equilateral triangle in half. What is the sum of the perimeters of $ACD$ and $BCD$?

A Quadnumber is a number whose digits sum to $4$. How many positive 3-digit quadnumbers are there? (ex. $310$ is a quadnumber because it is positive, 3-digit, and the digits sum to $4$)

A wheel has $100$ equal sections with the numbers from $1$ to $100$ written on them. If you spin the wheel $N$ times, what is the probability that you will land on a prime number on any of the $N$ spins? (Clarification: Probability of Landing on a prime AT LEAST ONCE in the N spins)

Car A travels at a constant $10$ mph. Car B is a probabilistic car, and at the start of every hour, it gets the option for a $10\%$ chance of traveling at $5$ mph, an $80\%$ chance of traveling at $10$ mph, and a $10\%$ chance of traveling at $15$ mph. Once Car B decides on a speed, it acts like a normal car for the rest of the hour with a constant speed at which it picked at the start of the hour, and then the same options happen at the start of the next hour, and so on. Car A and Car B leave at the same time, and they both go in the same direction. What is the probability that Car B is right next to Car A (same speed, same position) for only $3$ hours in the next 6 hours?

If $x \equiv 2\pmod{3}$, and $x \equiv 3\pmod{4}$, and $x \equiv 4\pmod{5}$, what is the least positive value of $x$.

If $f(x)=3x-1$ and $g(x)$ is $x^2-3x+3$, what is $f(g(5)-f(4))+g(f(3)-g(2))+1$?



This competition is shorter than the Mars version (MIMC 2200) which was 10 questions. But this one I believe is harder than the MIMC. (designed to be, anyway) Btw here is the entire MIMC competition in case you missed it a couple weeks ago.



$25^2-10^2=$?

If ball A costs $\$1.50$, and ball B costs $\$1.75$, what is the average cost per ball if you buy $10$ of ball A and $4$ of ball B? Please write as an improper fraction

How many divisors does $2200^3$ have?

If $f(2200-k)=10(k-2)$, then what is $f(f(2^{11}))$?

If three distinct positive primes, $(p,q,r)$, sum to 50, what is the greatest possible value of $(r^p)-(q^p)$ such that $p<q<r$?

What positive integer $x$ satisfies: $x^5-(x+4)^3+10^{(5-3)}=0$?

If a circle with radius $1$ is inscribed in a square, which is inscribed in another circle, what is the area of the outer circle minus the area of the inner circle?

What is the sum of $b$ and $c$ such that the roots of $x^2+bx+c=0$ sum to $11c$ and multiply to $12b-7$?

If the operation $a\#b = \dfrac{(a-b+1)^2}{(b-a+1)^2}$, what is the largest possible value of $(3\#x)$ given that $x$ is a positive integer?

How many 4-digit numbers have the property that the sum of the thousands and tens digits combined is the square of the sum of the hundreds and ones digits combined?

Pls write all solutions like this:
ex. S1 (Contest: either VIMC or MIMC)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Completed problems (I'm not going to show solutions here so other people can solve it)
MIMC: 1,2,3,4,5,6,7,8,9 (9/10 done)
VIMC: 1,3,6,7 (4/7 done)
45 replies
OWOW
Jul 16, 2025
OWOW
an hour ago
Divisibility
Justbrick   18
N an hour ago by huajun78
Prove that for all integers $n$.
Prove that \( n^5 - 5n^3 + 4n \) is divisible by 120
18 replies
Justbrick
Today at 1:35 AM
huajun78
an hour ago
Bonza functions
KevinYang2.71   61
N an hour ago by VulcanForge
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
61 replies
KevinYang2.71
Jul 15, 2025
VulcanForge
an hour ago
f(f(x)+y)=x+f(y) in R+
parmenides51   6
N an hour ago by Fly_into_the_sky
Source: 2014 Belarus TST 2.1
Find all functions$ f : R_+  \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$

(Folklore)

PS
6 replies
parmenides51
Dec 30, 2020
Fly_into_the_sky
an hour ago
AZE JBMO TST
IstekOlympiadTeam   6
N an hour ago by DensSv
Source: AZE JBMO TST
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]
6 replies
IstekOlympiadTeam
May 2, 2015
DensSv
an hour ago
IMO 2025 P2
sarjinius   86
N an hour ago by VulcanForge
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

Proposed by Trần Quang Hùng, Vietnam
86 replies
sarjinius
Jul 15, 2025
VulcanForge
an hour ago
Sunny lines
sarjinius   50
N an hour ago by VulcanForge
Source: 2025 IMO P1
A line in the plane is called $sunny$ if it is not parallel to any of the $x$axis, the $y$axis, or the line $x+y=0$.

Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
[list]
[*] for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
[*] exactly $k$ of the $n$ lines are sunny.
[/list]
Proposed by Linus Tang, USA
50 replies
sarjinius
Jul 15, 2025
VulcanForge
an hour ago
Random but useful theorems
booking   33
N 2 hours ago by gbatkhuu1
There have been all these random but useful theorems
Please post any theorems you know, random or not, but please say whether they are random or not.
I'll start give an example:
Random
I am just looking for some theorems to study.
33 replies
booking
Jul 16, 2025
gbatkhuu1
2 hours ago
I miss Turbo
sarjinius   33
N 2 hours ago by VulcanForge
Source: 2025 IMO P6
Consider a $2025\times2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.

Proposed by Zhao Yu Ma and David Lin Kewei, Singapore
33 replies
1 viewing
sarjinius
Jul 16, 2025
VulcanForge
2 hours ago
Hard sequence
straight   4
N 2 hours ago by straight
Source: Own
Consider a sequence $(a_n)_n, n \rightarrow \infty$ of real numbers.

Consider an infinite $\mathbb{N} \times \mathbb{N}$ grid $a_{i,j}$. In the first row of this grid, we place $a_0$ in every square ($a_{0,n} = a_0)$. In the first column of this grid, we place $a_n$ in the $n$-th square ($a_{n,0} = a_n)$.
Next, fill up the grid according to the following rule: $a_{i,j} = a_{i-1,j} + a_{i,j-1}$.

If $\lim_{i \rightarrow \infty} a_{i,j} = 0$ for all $j = 0,1,...$, does this mean that $a_n = 0$ for all $n$?

Hint?
4 replies
straight
Jul 16, 2025
straight
2 hours ago
2025 Greece IMO TST P3
brainfertilzer   13
N 2 hours ago by sami1618
Source: 2024 ISL G3
Let $ABCDE$ be a convex pentagon and let $M$ be the midpoint of $AB$. Suppose that segment $AB$ is tangent to the circumcircle of triangle $CME$ at $M$ and that $D$ lies on the circumircles of $AME$ and $BMC$. Lines $AD$ and $ME$ interesect at $K$, and lines $BD$ and $MC$ intersect at $L$. Points $P$ and $Q$ lie on line $EC$ so that $\angle PDC = \angle EDQ = \angle ADB$.

Prove that lines $KP, LQ,$ and $MD$ are concurrent.
13 replies
brainfertilzer
Jul 16, 2025
sami1618
2 hours ago
MAP Goals
Antoinette14   28
N May 18, 2025 by GallopingUnicorn45
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.
28 replies
Antoinette14
May 8, 2025
GallopingUnicorn45
May 18, 2025
MAP Goals
G H J
G H BBookmark kLocked kLocked NReply
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Antoinette14
38 posts
#1 • 2 Y
Y by Exponent11, PikaPika999
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.
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sadas123
1453 posts
#2 • 2 Y
Y by Exponent11, PikaPika999
I never took MAP but I guess my goal would be like 320+
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Inaaya
487 posts
#3 • 2 Y
Y by PikaPika999, CheckeredPenguin10
i was aiming for 300 but i got 295 :mad:
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Schintalpati
661 posts
#4
Y by
Antoinette14 wrote:
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.

350 next ig
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Antoinette14
38 posts
#5
Y by
Inaaya wrote:
i was aiming for 300 but i got 295 :mad:

rip
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anticodon
202 posts
#6
Y by
oops read title as "MOP"
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GallopingUnicorn45
546 posts
#7
Y by
am having map next week (science included) and i'm going to aim for 300
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Rice_Farmer
961 posts
#8
Y by
I have no enemies
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skronkmonster
2215 posts
#9
Y by
Schintalpati wrote:
Antoinette14 wrote:
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.

350 next ig

if it were possible for there to be a 120th percentile, AoPS students would be the sole achievers of it :rotfl:
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GallopingUnicorn45
546 posts
#10
Y by
totally lol
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Existing_Human1
218 posts
#11
Y by
What's map?
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Panda729
389 posts
#12 • 1 Y
Y by PikaPika999
Oh MAP is testing you on what you know and don’t to see what teachers need teach next time. :)
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GallopingUnicorn45
546 posts
#13 • 1 Y
Y by PikaPika999
in 2nd grade i had map testing and one of my classmates said she got 70 for math
i got 230-ish that time
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ZMB038
444 posts
#14 • 1 Y
Y by PikaPika999
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This post has been edited 1 time. Last edited by ZMB038, May 16, 2025, 9:05 PM
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Schintalpati
661 posts
#15
Y by
skronkmonster wrote:
Schintalpati wrote:
Antoinette14 wrote:
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.

350 next ig

if it were possible for there to be a 120th percentile, AoPS students would be the sole achievers of it :rotfl:

nou :crying: it would just be nice to achieve that from 336, I feel like I'll go down badly next year
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ZMB038
444 posts
#16
Y by
what is MAP?
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mhgelgi
864 posts
#17
Y by
search it up.
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Math-lover1
467 posts
#18
Y by
ZMB038 wrote:
what is MAP?

MAP stands for Measures of Academic Progress
basically a series of tests on math, reading, and other stuff (for school, not a math comp)
I don't think it's worth grinding though
This post has been edited 1 time. Last edited by Math-lover1, May 16, 2025, 10:03 PM
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Bocabulary142857
9 posts
#19
Y by
I got 304 on the Algebra I test
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tikachaudhuri
96 posts
#20
Y by
My highest was 302 in 7th grade winter, then 298 in 7th grade spring, and I didn't take it after that
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CJB19
178 posts
#21
Y by
Wait what is MAP
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mhgelgi
864 posts
#22
Y by
Math-lover1 wrote:
ZMB038 wrote:
what is MAP?

MAP stands for Measures of Academic Progress
basically a series of tests on math, reading, and other stuff (for school, not a math comp)
I don't think it's worth grinding though
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CJB19
178 posts
#23
Y by
ohhhh I see
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ZMB038
444 posts
#24
Y by
mhgelgi wrote:
Math-lover1 wrote:
ZMB038 wrote:
what is MAP?

MAP stands for Measures of Academic Progress
basically a series of tests on math, reading, and other stuff (for school, not a math comp)
I don't think it's worth grinding though

Thanks Math-Lover1
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deeptisidana
52 posts
#25 • 1 Y
Y by PikaPika999
Trust 300 by end of next year
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ImaadSiddiqui
6 posts
#26
Y by
200 is my goal
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hellohi321
113 posts
#27
Y by
ImaadSiddiqui wrote:
200 is my goal

A bit too high
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ZMB038
444 posts
#29
Y by
hellohi321 wrote:
ImaadSiddiqui wrote:
200 is my goal

A bit too high

:what?:
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GallopingUnicorn45
546 posts
#30
Y by
200 is lower than what I got in 2nd grade bro
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