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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.

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jlacosta
May 1, 2025
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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
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   16
N 9 minutes ago by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
16 replies
OgnjenTesic
May 22, 2025
JARP091
9 minutes ago
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equal segments on radiuses
danepale   8
N 13 minutes ago by zuat.e
Source: Croatia TST 2016
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.
8 replies
danepale
Apr 25, 2016
zuat.e
13 minutes ago
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Inequality
SunnyEvan   8
N 27 minutes ago by arqady
Let $a$, $b$, $c$ be non-negative real numbers, no two of which are zero. Prove that :
$$ \sum \frac{3ab-2bc+3ca}{3b^2+bc+3c^2} \geq \frac{12}{7}$$
8 replies
SunnyEvan
Apr 1, 2025
arqady
27 minutes ago
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Inequality conjecture
RainbowNeos   2
N 32 minutes ago by RainbowNeos
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
2 replies
RainbowNeos
May 29, 2025
RainbowNeos
32 minutes ago
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2- player game on a strip of n squares with two game pieces
parmenides51   2
N 33 minutes ago by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.

For which $n$ can Bob ensure a win no matter how Alice plays?
For which $n$ can Alice ensure a win no matter how Bob plays?

(Karl Czakler)
2 replies
parmenides51
Mar 26, 2024
Gggvds1
33 minutes ago
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Incenters and Circles
rkm0959   6
N 38 minutes ago by happypi31415
Source: Korean National Junior Olympiad Problem 1
In a triangle $\triangle ABC$ with incenter $I$,
Let $D$ = $AI$ $\cap$ $BC$
$E$ = incenter of $\triangle ABD$
$F$ = incenter of $\triangle ACD$
$P$ = intersection of $\odot BCE$ and $\overline {ED}$
$Q$ = intersection of $\odot BCF$ and $\overline {FD}$
$M$ = midpoint of $\overline {BC}$

Prove that $D, M, P, Q$ concycle
6 replies
rkm0959
Nov 2, 2014
happypi31415
38 minutes ago
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Reflected point lies on radical axis
Mahdi_Mashayekhi   6
N 38 minutes ago by khanhnx
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
6 replies
Mahdi_Mashayekhi
Apr 19, 2025
khanhnx
38 minutes ago
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Gcd of N and its coprime pair sum
EeEeRUT   19
N 41 minutes ago by HamstPan38825
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
19 replies
EeEeRUT
Apr 16, 2025
HamstPan38825
41 minutes ago
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Question on Balkan SL
Fmimch   4
N an hour ago by BreezeCrowd
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
4 replies
Fmimch
Apr 30, 2025
BreezeCrowd
an hour ago
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(3^{p-1} - 1)/p is a perfect square for prime p
parmenides51   4
N an hour ago by Rayvhs
Source: 2017 Saudi Arabia JBMO TST 1.2
Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.
4 replies
parmenides51
May 28, 2020
Rayvhs
an hour ago
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Rays, incircle, angles...
mathisreal   3
N an hour ago by Assassino9931
Source: Rioplatense L-3 2022 #4
Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.
3 replies
mathisreal
Dec 13, 2022
Assassino9931
an hour ago
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Find the value
sqing   0
2 hours ago
Source: Own
Let $ a,b $ be real numbers such that $ (a^2 + b^2) (a + 1) (b + 1) = a ^ 3 + b ^ 3 =2 $. Find the value of $ a b .$

Let $ a,b $ be real numbers such that $ (a^2 + b^2) (a + 1) (b + 1) = 2 $ and $ a ^ 3 + b ^ 3 = 1 $. Find the value of $ a + b .$
0 replies
sqing
2 hours ago
0 replies
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Wordy Geometry in Taiwan TST
ckliao914   9
N 2 hours ago by Scilyse
Source: 2023 Taiwan TST Round 3 Mock Exam 6
Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.
9 replies
ckliao914
Apr 29, 2023
Scilyse
2 hours ago
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Factorial Divisibility
Aryan-23   47
N 2 hours ago by ezpotd
Source: IMO SL 2022 N2
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
47 replies
Aryan-23
Jul 9, 2023
ezpotd
2 hours ago
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Irrational equation
giangtruong13   5
N Apr 25, 2025 by pooh123
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
5 replies
giangtruong13
Apr 24, 2025
pooh123
Apr 25, 2025
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Irrational equation
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giangtruong13
152 posts
giangtruong13
#1 h  Apr 24, 2025, 1:44 PM
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Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
This post has been edited 1 time. Last edited by giangtruong13, Apr 24, 2025, 1:44 PM
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Tuvshuu
11 posts
Tuvshuu
#2 h  Apr 24, 2025, 2:20 PM
Y by
$x = 4$.
I think it's only solution
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Tuvshuu
11 posts
Tuvshuu
#3 h  Apr 24, 2025, 2:37 PM
Y by
Let $f(x) = (\sqrt{x} + 1)[2 - (x - 6) \sqrt{x - 3}]$.
We can easily check $f''(x) \leq 0$ then $f$ is concave function.
$x + 8$ is the $f$'s tangent on the $x = 4$ then only solution is $x = 4$ ($f'(4) = 1$ and $f(4) = 12$).
Attachments:
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navier3072
121 posts
navier3072
#4 h  Apr 24, 2025, 2:58 PM
Y by
How did you show $f''(x)\leq 0$? It seems like a very complicated function
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giangtruong13
152 posts
giangtruong13
#5 h  Apr 25, 2025, 8:19 AM
Y by
boom chalaka
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pooh123
81 posts
pooh123
#6 h  Apr 25, 2025, 1:31 PM
Y by
giangtruong13 wrote:
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$

The domain of the equation is \( x \geq 3 \). We have:

\[ (\sqrt{x} + 1)\left[2 - (x-6)\sqrt{x-3}\right] = x + 8 \]
\[ \Leftrightarrow x + 8 + (\sqrt{x} + 1)\left[(x-6)\sqrt{x-3} - 2\right] = 0 \]
\[ \Leftrightarrow x + 8 - 4\sqrt{x} - 4 + (\sqrt{x} + 1)\left[(x-6)\sqrt{x-3} + 2\right] = 0 \]
\[ \Leftrightarrow (\sqrt{x} - 2)^2 + (\sqrt{x} + 1)\left[(x-3)\sqrt{x-3} - 3\sqrt{x-3} + 2\right] = 0 \]
\[ \Leftrightarrow (\sqrt{x} - 2)^2 + (\sqrt{x} + 1)(\sqrt{x-3}+2)(\sqrt{x-3}-1)^2 = 0 \]
Since each term on the left-hand side is non-negative, the left-hand side must be non-negative.
Therefore, \(\sqrt{x} - 2 = 0\) and \(\sqrt{x-3} - 1 = 0\), so \(x = 4\), which is in the domain.

Hence the only solution to the equation is \(x = 4\).
This post has been edited 2 times. Last edited by pooh123, Apr 25, 2025, 1:42 PM
Reason: typo
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