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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)!

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[*]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
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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
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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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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Tangent circles
Vlados021   3
N a minute ago by fearsum_fyz
Source: 2017 Belarus Team Selection Test 6.3
Given an isosceles triangle $ABC$ with $AB=AC$. let $\omega(XYZ)$ be the circumcircle of a triangle $XYZ$. Tangents to $\omega(ABC)$ at $B$ and $C$ meet at $D$. Point $F$ is marked on the arc $AB$ (opposite to $C$). Let $K$, $L$ be the intersection points of $AF$ and $BD$, $AB$ and $CF$, respectively.
Prove that if circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other, the their tangency point belongs to $AB$. (Here $T$ and $S$ are the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively.)
3 replies
Vlados021
Mar 31, 2019
fearsum_fyz
a minute ago
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Inspired by SunnyEvan
sqing   0
10 minutes ago
Source: Own
Let $ a,b,c $ be reals such that $ a^2+b^2+c^2=5(ab+bc+ca). $ Prove that$$ -\frac{1}{25} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{197}{675} $$
0 replies
1 viewing
sqing
10 minutes ago
0 replies
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3 var inequality
SunnyEvan   1
N 13 minutes ago by sqing
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{53}{2}-9\sqrt{14} \leq \frac{8(a^3b+b^3c+c^3a)}{27(a^2+b^2+c^2)^2} \leq \frac{53}{2}+9\sqrt{14} $$
1 reply
SunnyEvan
2 hours ago
sqing
13 minutes ago
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Prove that the triangle is isosceles.
TUAN2k8   5
N 17 minutes ago by JARP091
Source: My book
Given acute triangle $ABC$ with two altitudes $CF$ and $BE$.Let $D$ be the point on the line $CF$ such that $DB \perp BC$.The lines $AD$ and $EF$ intersect at point $X$, and $Y$ is the point on segment $BX$ such that $CY \perp BY$.Suppose that $CF$ bisects $BE$.Prove that triangle $ACY$ is isosceles.
5 replies
TUAN2k8
Yesterday at 9:55 AM
JARP091
17 minutes ago
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one variable function
youochange   0
20 minutes ago
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
0 replies
+1 w
youochange
20 minutes ago
0 replies
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Israel Number Theory
mathisreaI   64
N 21 minutes ago by Jlzh25
Source: IMO 2022 Problem 5
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
64 replies
mathisreaI
Jul 13, 2022
Jlzh25
21 minutes ago
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Find the minimum
sqing   7
N 28 minutes ago by sqing
Source: China Shandong High School Mathematics Competition 2025 Q4
Let $ a,b,c>0,abc>1$. Find the minimum value of $ \frac {abc(a+b+c+8)}{abc-1}. $
7 replies
sqing
6 hours ago
sqing
28 minutes ago
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Insspired by Shandong 2025
sqing   5
N 32 minutes ago by sqing
Source: Own
Let $ a,b,c>0,abc>1$. Prove that$$ \frac {abc(a+b+c+ab+bc+ca+3)}{ abc-1}\geq \frac {81}{4}$$$$ \frac {abc(a+b+c+ab+bc+ca+abc+2)}{ abc-1}\geq 12+8\sqrt{2}$$
5 replies
sqing
6 hours ago
sqing
32 minutes ago
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Bound of number of connected components
a_507_bc   3
N 2 hours ago by MmdMathLover
Source: St. Petersburg 2023 11.7
Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?
3 replies
a_507_bc
Aug 12, 2023
MmdMathLover
2 hours ago
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A circle tangent to the circumcircle, excircles related
kosmonauten3114   0
2 hours ago
Source: My own, maybe well-known
Let $ABC$ be a scalene triangle with excircles $\odot(I_A)$, $\odot(I_B)$, $\odot(I_C)$. Let $\odot(A')$ be the circle which touches $\odot(I_B)$ and $\odot(I_C)$ and passes through $A$, and whose center $A'$ lies outside of the excentral triangle of $\triangle{ABC}$. Define $\odot(B')$ and $\odot(C')$ cyclically. Let $\odot(O')$ be the circle externally tangent to $\odot(A')$, $\odot(B')$, $\odot(C')$.

Prove that $\odot(O')$ is tangent to the circumcircle of $\triangle{ABC}$ at the anticomplement of the Feuerbach point of $\triangle{ABC}$.
0 replies
kosmonauten3114
2 hours ago
0 replies
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Bounds on degree of polynomials
Phorphyrion   4
N 2 hours ago by Kingsbane2139
Source: 2020 Israel Olympic Revenge P3
For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds:

For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, all of the same size, let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which
\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]
a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$.

b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.
4 replies
Phorphyrion
Jun 11, 2022
Kingsbane2139
2 hours ago
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A point on BC
jayme   7
N 3 hours ago by jayme
Source: Own ?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.

Prove : B, C and T are collinear.

Sincerely
Jean-Louis
7 replies
jayme
Today at 6:08 AM
jayme
3 hours ago
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Zack likes Moving Points
pinetree1   73
N 3 hours ago by NumberzAndStuff
Source: USA TSTST 2019 Problem 5
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$.

Gunmay Handa
73 replies
pinetree1
Jun 25, 2019
NumberzAndStuff
3 hours ago
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Domain and Inequality
Kunihiko_Chikaya   1
N 3 hours ago by Mathzeus1024
Source: 2018 The University of Tokyo entrance exam / Humanities, Problem 1
Define on a coordinate plane, the parabola $C:y=x^2-3x+4$ and the domain $D:y\geq x^2-3x+4.$
Suppose that two lines $l,\ m$ passing through the origin touch $C$.

(1) Let $A$ be a mobile point on the parabola $C$. Let denote $L,\ M$ the distances between the point $A$ and the lines $l,\ m$ respectively. Find the coordinate of the point $A$ giving the minimum value of $\sqrt{L}+\sqrt{M}.$

(2) Draw the domain of the set of the points $P(p,\ q)$ on a coordinate plane such that for all points $(x,\ y)$ over the domain $D$, the inequality $px+qy\leq 0$ holds.
1 reply
Kunihiko_Chikaya
Feb 25, 2018
Mathzeus1024
3 hours ago
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Gen func 8s somewhere in the balkans
Assassino9931   1
N Apr 27, 2025 by MR.1
Source: Balkan MO 2024 Shortlist A2
Prove that there is a positive integer such that the decimal representation of
$$ \sum_{k=1}^{\lfloor n/3 \rfloor} \binom{n}{3k}8^k $$has all of its last $2023$ digits equal to $8$.
1 reply
Assassino9931
Apr 27, 2025
MR.1
Apr 27, 2025
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Gen func 8s somewhere in the balkans
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Reveal topic tags
algebra
generating functions
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Source: Balkan MO 2024 Shortlist A2
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Assassino9931
1354 posts
Assassino9931
#1 h  Apr 27, 2025, 12:38 PM
Y by
Prove that there is a positive integer such that the decimal representation of
$$ \sum_{k=1}^{\lfloor n/3 \rfloor} \binom{n}{3k}8^k $$has all of its last $2023$ digits equal to $8$.
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MR.1
128 posts
MR.1
#2 h  Apr 27, 2025, 12:41 PM
Y by
also georgia bmo tst p2
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