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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
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Summer camps are starting this 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 a transformative 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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0 replies
jlacosta
Jun 2, 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
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
J
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tangent circles
m4thbl3nd3r   0
a few seconds ago
Let $O,H,T$ be circumcenter, orthocenter and A-HM point of triangle $ABC$. Let $AH,AT$ intersect $(O)$ at $K,N$, respectively. Let $XYZ$ be the triangle formed by $TH,BC,KN$. Prove that $(XYZ)$ is tangent to $(O).$
0 replies
m4thbl3nd3r
a few seconds ago
0 replies
J
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Brilliant Problem
M11100111001Y1R   9
N 12 minutes ago by The5
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[ \frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2 \]
9 replies
M11100111001Y1R
May 27, 2025
The5
12 minutes ago
J
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Draw sqrt(2024)
shanelin-sigma   1
N 20 minutes ago by CrazyInMath
Source: 2024/12/24 TCFMSG Mock p10
On a big plane, two points with length $1$ are given. Prove that one can only use straightedge (which draws a straight line passing two drawn points) and compass (which draws a circle with a chosen radius equal to the distance of two drawn points and centered at a drawn points) to construct a line and two points on it with length $\sqrt{2024}$ in only $10$ steps (Namely, the total number of circles and straight lines drawn is at most $10$.)
1 reply
shanelin-sigma
Dec 24, 2024
CrazyInMath
20 minutes ago
J
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A beautiful Lemoine point problem
phonghatemath   3
N 27 minutes ago by orengo42
Source: my teacher
Given triangle $ABC$ inscribed in a circle with center $O$. $P$ is any point not on (O). $AP, BP, CP$ intersect $(O)$ at $A', B', C'$. Let $L, L'$ be the Lemoine points of triangle $ABC, A'B'C'$ respectively. Prove that $P, L, L'$ are collinear.
3 replies
phonghatemath
Today at 5:01 AM
orengo42
27 minutes ago
J
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Serbian selection contest for the IMO 2025 - P1
OgnjenTesic   4
N 38 minutes ago by Mathgloggers
Source: Serbian selection contest for the IMO 2025
Let \( p \geq 7 \) be a prime number and \( m \in \mathbb{N} \). Prove that
\[\left| p^m - (p - 2)! \right| > p^2.\]Proposed by Miloš Milićev
4 replies
OgnjenTesic
May 22, 2025
Mathgloggers
38 minutes ago
J
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Complex number
soruz   1
N 43 minutes ago by Mathzeus1024
$i)$ Determine $z \in \mathbb{C} $ such that $2|z| \ge |z^n-3|, \forall n \in \mathbb N^*.$
$ii)$ Determine $z \in \mathbb{C} $ such that $2|z| \ge |z^n+3|, \forall n \in \mathbb N^*.$
1 reply
soruz
Nov 28, 2024
Mathzeus1024
43 minutes ago
J
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k colorings and triangles
Rijul saini   2
N an hour ago by kotmhn
Source: LMAO Revenge 2025 Day 1 Problem 3
In the city of Timbuktu, there is an orphanage. It shelters children from the new mysterious disease that causes children to explode. There are m children in the orphanage. To try to cure this disease, a mad scientist named Myla has come up with an innovative cure. She ties every child to every other child using medicinal ropes. Every child is connected to every other child using one of $k$ different ropes. She then performs a experiment that causes $3$ children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.

We let $f(n, k)$ be the minimum m such that Myla can perform at least $n$ experiments of the same type. Prove that:

$i.$ For every $k \in \mathbb N$ there exists a $N_k \in N$ and $a_k, b_k \in \mathbb Z$ such that for all $n > N_k$, \[f(n, k) = a_kn + b_k.\]
$ii.$ Find the value of $a_k$ for every $k \in \mathbb N$.
2 replies
Rijul saini
Wednesday at 7:11 PM
kotmhn
an hour ago
J
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IMO ShortList 2008, Number Theory problem 1
April   65
N an hour ago by Siddharthmaybe
Source: IMO ShortList 2008, Number Theory problem 1
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.

Proposed by Angelo Di Pasquale, Australia
65 replies
April
Jul 9, 2009
Siddharthmaybe
an hour ago
J
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Aloo and Batata play game on N-gon
guptaamitu1   0
an hour ago
Source: LMAO revenge 2025 P6
Aloo and Batata are playing a game. They are given a regular $n$-gon, where $n > 2$ is an even integer. At the start, a line joining two vertices is chosen arbitrarily and one of its endpoints is chosen as its pivot. Now, Aloo rotates the line around the pivot either clockwise or anti-clockwise until it passes through another vertex of the $n$-gon. Then, the new vertex becomes the pivot and Batata again chooses to rotate the line clockwise or anti-clockwise
about the pivot. The player who moves the line into a position it has already been in (i.e. it passes through the same two vertices of the $n$-gon it was in at a previous time) loses.
Find all $n$ such that Batata always has a winning strategy irrespective of the starting edge.

Proposed by Anik Sardar, Om Patil and Anudip Giri
0 replies
guptaamitu1
an hour ago
0 replies
J
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Trig Inequality back in Olympiads!
guptaamitu1   0
an hour ago
Source: LMAO revenge 2025 P5
Let $x,y,z \in \mathbb R$ be such that $x + y + z = \frac{\pi}{2}$ and $0 < x,y,z \le \frac{\pi}{4}$. Prove that:
$$ \left( \frac{\sin x - \sin y}{\cos z} \right)^2 \le 1 - 8 \sin x \sin y \sin z $$
Proposed by Shreyas Deshpande
0 replies
guptaamitu1
an hour ago
0 replies
J
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Reflection of (BHC) in AH
guptaamitu1   0
an hour ago
Source: LMAO revenge 2025 P4
Let $ABC$ be a triangle with orthocentre $H$. Let $D,E,F$ be the foot of altitudes of $A,B,C$ onto the opposite sides, respectively. Consider $\omega$, the reflection of $\odot(BHC)$ about line $AH$. Let line $EF$ cut $\omega$ at distinct points $X,Y$, and let $H'$ be the orthocenter of $\triangle AYD$. Prove that points $A,H',X,D$ are concyclic.

Proposed by Mandar Kasulkar
0 replies
guptaamitu1
an hour ago
0 replies
J
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King's Constrained Walk
Hellowings   2
N an hour ago by Hellowings
Source: Own
Given an n x n chessboard, with a king starting at any square, the king's task is to visit each square in the board exactly once (essentially an open path); this king moves how a king in chess would.
However, we are allowed to place k numbers on the board of any value such that for each number A we placed on the board, the king must be in the position of that number A on its Ath square in its journey, with the starting square as its 1st square.
Suppose after we placed k numbers, there is one and only one way to complete the king's task (this includes placing the king in a starting square), find the minimum value of k set by n.

Should've put one of its tag as "Open problem"; I have no idea how to tackle this problem either.
2 replies
Hellowings
May 30, 2025
Hellowings
an hour ago
J
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Nut equation
giangtruong13   2
N 2 hours ago by Mathzeus1024
Source: Mie black fiends
Solve the quadratic equation: $$[4(\sqrt{(1+x)^3})^2-3\sqrt{1+x^2}](4x^3+3x)=2$$
2 replies
giangtruong13
Apr 1, 2025
Mathzeus1024
2 hours ago
J
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Euler line problem
m4thbl3nd3r   2
N 2 hours ago by m4thbl3nd3r
Let $O,H$ be the circumcenter and orthocenter of triangle $ABC$ and $E,F$ be intersections of $OH$ with $AB,AC$. Let $H',O'$ be orthocenter and circumcenter of triangle $AEF$. Prove that $O'H'\parallel BC.$
2 replies
m4thbl3nd3r
3 hours ago
m4thbl3nd3r
2 hours ago
J
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J
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JBMO Shortlist 2020 N1
Lukaluce   7
N Apr 12, 2025 by MATHS_ENTUSIAST
Source: JBMO Shortlist 2020
Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.
7 replies
Lukaluce
Jul 4, 2021
MATHS_ENTUSIAST
Apr 12, 2025
J
JBMO Shortlist 2020 N1
G H J
Reveal topic tags
number theory
JBMO Shortlist
AZE JUNIOR NMO
L
G H BBookmark kLocked kLocked NReply
Source: JBMO Shortlist 2020
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Lukaluce
274 posts
Lukaluce
#1 h  Jul 4, 2021, 2:47 PM • 2 Y
Y by Stepinac, Sedro
Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.
Z K Y
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math31415926535
5617 posts
math31415926535
#2 h  Jul 4, 2021, 2:49 PM
Y by
https://artofproblemsolving.com/community/c6h2609453_8n47_is_prime
Z K Y
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Lukaluce
274 posts
Lukaluce
#4 h  Jul 4, 2021, 3:20 PM
Y by
This problem was proposed by Cyprus
Z K Y
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KhayalAliyev
7 posts
KhayalAliyev
#5 h  May 8, 2022, 12:44 PM • 2 Y
Y by Iora, Math_legendno12
If n=4k,4k+2 then this expression is equal 0 mod 3.If n=4k+1,then expression is 0 mod 5.If n=4k+3 then expression is 0 mod 13.I think this is very easy problem for JBMO and there is nothing difficult to understand
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Rodymaths
4 posts
Rodymaths
#8 h  May 6, 2024, 6:18 PM
Y by
From mod 3 we get that n should be odd (2k+1) and from mod 5 we get that n=4k+1 is not valid (then 8^n +47 will be divisible by 5) so the only solution has to be n=4k+3, but 8^(4k+3) is 5 mod 13 and 47 is -5 mod 13, so then 8^n+47 will be divisible by 13. In conclusion there are no solutions
Z K Y
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Sedro
5858 posts
Sedro
#9 h  May 11, 2024, 11:37 PM
Y by
No. Note that $8^n+47\equiv 2^n + 2\pmod{3}$, so we must have $n$ odd. Also, we have $8^n+47\equiv 3^n+2\pmod{5}$, and hence $n$ cannot be equivalent to $1$ modulo $4$. Let $n = 4n_0+3$. Then, $8^n+47\equiv 8^{4n_0}\cdot8^3 -5 \equiv 512-5\equiv 0 \pmod{13}$. Since $8^n+47>3,5,13$, we are done.
Z K Y
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MR.1
138 posts
MR.1
#10 h  Feb 16, 2025, 7:05 AM • 1 Y
Y by MATHS_ENTUSIAST
$\pmod{3}$ $\pmod{5}$ and $\pmod{13}$
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MATHS_ENTUSIAST
28 posts
MATHS_ENTUSIAST
#11 h  Apr 12, 2025, 10:23 AM
Y by
CASE 1 :n=even,Let n=2m
8^2m + 47=prime
64^m +47=prime
64^m≡1(mod3)
64^m+47≡48≡0(mod3)
No solⁿ
CASE 2 :n=odd, Let n=2m+1
8^2m+1 + 47=prime
64^m·8 + 47=prime

a] m=odd
64^m≡-1(mod13)
64^m·8≡-8(mod13)
64^m·8+47≡39≡0(mod13)
No solⁿ

b]m=even,Let m=2k
8^4k·8+47=prime
BY FLT
8^4≡1(mod5)
8^4k≡1(mod5)
8^4k·8≡8(mod5)
8^4k·8+47≡55≡0(mod5)
No solⁿ

ANS:NO SOLUTION
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