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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Yesterday at 3:18 PM
0 replies
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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
1 viewing
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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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
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2025 USAMO Rubric
plang2008   18
N 4 minutes ago by mathprodigy2011
1. Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

Rubric for Problem 1

2. Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Rubric for Problem 2

3. Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds:
[center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center]
[center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center]
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.

Note: $\triangle UVW$ is directly similar to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Rubric for Problem 3

4. Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Rubric for Problem 4

5. Determine, with proof, all positive integers $k$ such that \[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]is an integer for every positive integer $n$.

Rubric for Problem 5

6. Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.

Rubric for Problem 6
18 replies
plang2008
Yesterday at 1:33 AM
mathprodigy2011
4 minutes ago
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9 Can I make MOP
Bigtree   30
N 17 minutes ago by babyzombievillager
My dream is to be on IMO team ik thats not going to happen b/c the kids that make it are like 6th mop quals :play_ball:. I somehow got a $208.5$ index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs. made mc nats
30 replies
Bigtree
Mar 9, 2025
babyzombievillager
17 minutes ago
J
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9 Which math contest is your favorite?
mdk2013   59
N 35 minutes ago by rhydon516
links for details on each comp

i think thats all the major ones, comment if you have any more that i forgot to include

upvote if you're like me and you think MATHCOUNTS is obv better!

EDIT: i forgot ARML and JBMO, comment if you like those
59 replies
mdk2013
Mar 30, 2025
rhydon516
35 minutes ago
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A Confused Canadian
sximoz   3
N 36 minutes ago by Gavin_Deng
I always wanted to apply for the AMC, and this year, I think I might have a chance. A friend did AMC 8, and she came back telling me I should do it too. I was really enthusiastic, and wanted to apply.

I do not have prior experience with the AMC, and I live in Alberta, Canada. Through my research, I learned that applications must be submitted via an International Group Leader. However, I am uncertain about who they are and what steps I need to take in order to apply. If you have any information about the application, I would greatly appreciate your help.

Additionally, I would be grateful for any advice on how to best prepare for the AMC, particularly the AMC 8, as this may be one of my last opportunities to participate before moving on to the AMC 10. Specifically, I am interested in understanding the scoring system, the format of the contest, and whether it is possible to participate online from my location.

If you have any further info or tips, I would sincerely appreciate your assistance.

Thanks you very much,
sximoz
3 replies
+1 w
sximoz
an hour ago
Gavin_Deng
36 minutes ago
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Goes through fixed points
CheshireOrb   5
N 2 hours ago by HoRI_DA_GRe8
Source: Vietnam TST 2021 P5
Given a fixed circle $(O)$ and two fixed points $B, C$ on that circle, let $A$ be a moving point on $(O)$ such that $\triangle ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$ and let $AD, BE, CF$ be the three heights of $\triangle ABC$. In two rays $\overrightarrow{FA}, \overrightarrow{EA}$, we pick respectively $M,N$ such that $FM = CE, EN = BF$. Let $L$ be the intersection of $MN$ and $EF$, and let $G \neq L$ be the second intersection of $(LEN)$ and $(LFM)$.

a) Show that the circle $(MNG)$ always goes through a fixed point.

b) Let $AD$ intersects $(O)$ at $K \neq A$. In the tangent line through $D$ of $(DKI)$, we pick $P,Q$ such that $GP \parallel AB, GQ \parallel AC$. Let $T$ be the center of $(GPQ)$. Show that $GT$ always goes through a fixed point.
5 replies
CheshireOrb
Apr 2, 2021
HoRI_DA_GRe8
2 hours ago
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Unsolved NT, 3rd time posting
GreekIdiot   8
N 3 hours ago by ektorasmiliotis
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
8 replies
GreekIdiot
Mar 26, 2025
ektorasmiliotis
3 hours ago
J
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n=y^2+108
Havu   6
N 3 hours ago by ektorasmiliotis
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
6 replies
Havu
Today at 4:30 PM
ektorasmiliotis
3 hours ago
J
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Valuable subsets of segments in [1;n]
NO_SQUARES   0
3 hours ago
Source: Russian May TST to IMO 2023; group of candidates P6; group of non-candidates P8
The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
0 replies
NO_SQUARES
3 hours ago
0 replies
J
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Fneqn or Realpoly?
Mathandski   2
N 4 hours ago by jasperE3
Source: India, not sure which year. Found in OTIS pset
Find all polynomials $P$ with real coefficients obeying
\[P(x) P(x+1) = P(x^2 + x + 1)\]for all real numbers $x$.
2 replies
Mathandski
6 hours ago
jasperE3
4 hours ago
J
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thanks u!
Ruji2018252   2
N 4 hours ago by CHESSR1DER
find all $f: \mathbb{R}\to \mathbb{R}$ and
\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]
2 replies
Ruji2018252
6 hours ago
CHESSR1DER
4 hours ago
J
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Functional equations
hanzo.ei   11
N 6 hours ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
11 replies
hanzo.ei
Mar 29, 2025
GreekIdiot
6 hours ago
J
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D1018 : Can you do that ?
Dattier   1
N 6 hours ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
6 hours ago
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D1010 : How it is possible ?
Dattier   14
N Today at 5:30 PM by ehuseyinyigit
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
14 replies
Dattier
Mar 10, 2025
ehuseyinyigit
Today at 5:30 PM
J
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iran tst 2018 geometry
Etemadi   10
N Today at 5:05 PM by amirhsz
Source: Iranian TST 2018, second exam day 2, problem 5
Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.

Proposed by Ali Zamani, Hooman Fattahi
10 replies
Etemadi
Apr 17, 2018
amirhsz
Today at 5:05 PM
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J
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Colored Pencils for Math Competitions
Owinner   17
N Apr 1, 2025 by lord_of_the_rook
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.
17 replies
Owinner
Mar 29, 2025
lord_of_the_rook
Apr 1, 2025
J
Colored Pencils for Math Competitions
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Reveal topic tags
geometry
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Owinner
53 posts
Owinner
#1 h  Mar 29, 2025, 5:56 PM
Y by
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.
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mathprodigy2011
248 posts
mathprodigy2011
#2 h  Mar 29, 2025, 6:49 PM
Y by
Owinner wrote:
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.

i think colored pencils are more useful for really complicated and involved geometry problems. So I would think they are useful for AMC 10 but mathcounts and amc8 usually have geometry problems with simple-ish solutions
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LearnMath_105
139 posts
LearnMath_105
#3 h  Mar 29, 2025, 7:19 PM
Y by
I would say that they are most helpful for marking concyclic/colinear/concurrent objects so later aime - olyis where its most commonly used
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Andyluo
893 posts
Andyluo
#4 h  Mar 29, 2025, 7:43 PM
Y by
they're useless for the competitions listed, it's way too fast-paced to spend time and color a diagram, maybe on olympiads but you're losing precious time if you do it on those. Maybe the AIME would utilize this? But I highly doubt that it would give any benefits

@below coloring lines is basically the same thing c'mon
This post has been edited 1 time. Last edited by Andyluo, Mar 29, 2025, 7:49 PM
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cheltstudent
548 posts
cheltstudent
#5 h  Mar 29, 2025, 7:48 PM
Y by
Bruh, andyluo, literally not colouring just marking different lines they talked about this in my AMC 10 seminar

@above nah like drawing lines with a colour pencil bro
This post has been edited 1 time. Last edited by cheltstudent, Mar 29, 2025, 7:51 PM
Reason: gg
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ChaitraliKA
1003 posts
ChaitraliKA
#6 h  Mar 29, 2025, 7:57 PM
Y by
It can be useful, but I feel like in the actual competition you would end up not using it because you would be too invested in thinking about how to solve the problem instead of "oh, I should mark this with a different color", and would just end up drawing over your diagram with your regular pencil again. Drawing a specific part of your diagram with harder pressure compared to other areas of your diagram can also help you too, especially if it's just mathcounts or AMC 8.
I brought a ruler, a pen, and a compass to Aime, but didn't use any of those materials. Didn't touch them at all. No time to think ¯⁠\⁠_⁠(⁠ツ⁠)⁠_⁠/⁠¯
The only time I've ever used different colors is when I draw my diagrams on my chromebook canvas, and I'm lazy to get paper and pencil.
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mathprodigy2011
248 posts
mathprodigy2011
#7 h  Mar 29, 2025, 10:45 PM
Y by
Andyluo wrote:
they're useless for the competitions listed, it's way too fast-paced to spend time and color a diagram, maybe on olympiads but you're losing precious time if you do it on those. Maybe the AIME would utilize this? But I highly doubt that it would give any benefits

@below coloring lines is basically the same thing c'mon

i mean i used different colors to get aime p14 correct so they are useful just not for faster paced competitions. It really depends on how you think.
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mdk2013
562 posts
mdk2013
#8 h  Mar 29, 2025, 11:38 PM • 1 Y
Y by Exponent11
colored pencils way better for olympiads so you can show your steps, for example, there were a few problems on the BAMO and SDMO that i took that i could have used colored pencils to more accurately show my proofs. however
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Jack_w
108 posts
Jack_w
#9 h  Mar 30, 2025, 5:52 PM
Y by
it’s not very helpful
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resources
748 posts
resources
#10 h  Mar 30, 2025, 5:56 PM
Y by
they can be extremely helpful if used correctly.
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vincentwant
1283 posts
vincentwant
#11 h  Mar 30, 2025, 6:31 PM
Y by
I used colored pencils on jmo p3
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EaZ_Shadow
1155 posts
EaZ_Shadow
#12 h  Mar 30, 2025, 6:37 PM
Y by
Owinner wrote:
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.

Ofc u can use it for those comp if u want
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ChaitraliKA
1003 posts
ChaitraliKA
#13 h  Mar 30, 2025, 7:13 PM
Y by
@op, you're getting a bunch of different responses, but in the end, it's your decision obv, whether you wanna use it or not
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Andyluo
893 posts
Andyluo
#14 h  Mar 30, 2025, 7:23 PM
Y by
the general consensus is using colored pencils is a waste of time unless you're doing AIME/Olympiads, at that point, you choose if you should or should not.
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BS2012
970 posts
BS2012
#15 h  Mar 30, 2025, 7:28 PM
Y by
for contests AIME difficulty or easier the geo diagrams are usually simple enough that using colored pencils would be useless, for oly could be useful though
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Mr.Sharkman
496 posts
Mr.Sharkman
#16 h  Mar 31, 2025, 8:16 PM
Y by
vincentwant wrote:
I used colored pencils on jmo p3

Lol same; it kinda helped solve it
I also did on #2 of last year but still got it wrong anyway
This post has been edited 1 time. Last edited by Mr.Sharkman, Mar 31, 2025, 8:17 PM
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deduck
186 posts
deduck
#17 h  Mar 31, 2025, 9:48 PM • 1 Y
Y by fake123
colored pencils make me confused but i think on olympiads it's personal preference

i think on mathcounts/amc8/10/12 and other fast paced comeptition it's not worth it. if u cant solve the problem in pencil black/white it probably wont get solved just bc u colored it xd because usually the solutions arent that invovled about concyclic/collinear/etc

just my opinion xd
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lord_of_the_rook
130 posts
lord_of_the_rook
#18 h  Apr 1, 2025, 1:33 AM
Y by
@op I feel like you should try them out at home, and see if you find them useful.
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