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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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[*]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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Inspired by RMO 2006
sqing   2
N a minute ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1} $$Where $k\geq 0.03 $
$$ \frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3 $$
2 replies
sqing
Yesterday at 3:24 PM
sqing
a minute ago
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Inspired by 2025 Beijing
sqing   3
N 3 minutes ago by sqing
Source: Own
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
3 replies
sqing
Yesterday at 4:56 PM
sqing
3 minutes ago
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Inspired by 2025 Xinjiang
sqing   3
N 7 minutes ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(6+\frac {b}{a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq\frac {625}{ 12}$$$$ \left(1+\frac {a} { b}\right)\left(6+\frac {a}{b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq29+6\sqrt 6$$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {b}{ a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{3(63+11\sqrt{33})}{16} $$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{223+70\sqrt{10}}{27} $$
3 replies
1 viewing
sqing
Yesterday at 5:56 PM
sqing
7 minutes ago
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interesting diophantiic fe in natural numbers
skellyrah   2
N 20 minutes ago by SYBARUPEMULA
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m, n \in \mathbb{N} \),
\[ mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!). \]
2 replies
skellyrah
Yesterday at 8:01 AM
SYBARUPEMULA
20 minutes ago
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PHP on subsets
SYBARUPEMULA   0
an hour ago
Source: inspired by Romania 2009
Let $X$ be a set of $102$ elements. Let $A_1, A_2, A_3, ..., A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $100$ elements. Show that there's a member of $X$ that occurs in at least $7$ different subsets $A_j$.
0 replies
SYBARUPEMULA
an hour ago
0 replies
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Moscow Geometry Problems
nataliaonline75   1
N an hour ago by MathLuis
Source: MMO 2003 10.4
Let M be the intersection point of the medians of ABC. On the perpendiculars dropped from M to sides BC, AC, AB, points A1, B1, C1 are taken, respectively, with A1B1 perpendicular to MC and A1C1 perpendicular to MB. prove that M is the intersections pf the medians in A1B1C1.
Any solutions without vectors? :)
1 reply
1 viewing
nataliaonline75
Jul 9, 2024
MathLuis
an hour ago
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Serbian selection contest for the IMO 2025 - P4
OgnjenTesic   1
N an hour ago by nataliaonline75
Source: Serbian selection contest for the IMO 2025
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its colorfulness as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović
1 reply
1 viewing
OgnjenTesic
May 22, 2025
nataliaonline75
an hour ago
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Another OI geometry problem
chengbilly   7
N an hour ago by MathLuis
Source: own
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. $H$ the orthocenter of triangle $BIC$ . The incircle of $ABC$ touches side $AC,AB$ at $E,F$ respectively. Suppose that $\odot (AEF)$ and $\odot (AIO)$ intersects $\odot (ABC)$ at $S$ and $T$ respectively (differ from $A$). Prove that $T,H,I,S$ lies on a circle.

($\odot (ABC)$ denote the circumcircle of $ABC$)
7 replies
chengbilly
Apr 14, 2021
MathLuis
an hour ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   2
N an hour ago by nataliaonline75
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
2 replies
OgnjenTesic
May 22, 2025
nataliaonline75
an hour ago
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Infinite Pairs of Integers
steven_zhang123   0
2 hours ago
Source: 2025 Spring NSMO P6
Given a positive integer \( k \), prove that there exist infinitely many pairs of positive integers \((m, n)\) (\(m < n\)) such that
\[ \tau(m^k) \tau(m^k + 1) \cdots \tau(n^k - 1) \tau(n^k) \]is a perfect square, where \(\tau(n)\) denotes the number of positive divisors of \(n\).
Proposed by Dong Zhenyu
0 replies
steven_zhang123
2 hours ago
0 replies
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power sum system of equations in 3 variables
Stear14   0
2 hours ago
Given that
$x^2+y^2+z^2=8\ ,$
$x^3+y^3+z^3=15\ ,$
$x^5+y^5+z^5=100\ .$

Find the value of $\ x+y+z\ .$
0 replies
Stear14
2 hours ago
0 replies
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Game on 6 by 6 grid
billzhao   26
N 2 hours ago by Sleepy_Head
Source: USAMO 2004, problem 4
Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
26 replies
billzhao
Apr 29, 2004
Sleepy_Head
2 hours ago
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USA GEO 2003
dreammath   21
N 3 hours ago by lpieleanu
Source: TST USA 2003
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that
\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \]
if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)
21 replies
dreammath
Feb 16, 2004
lpieleanu
3 hours ago
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Balkan Mathematical Olympiad
ABCD1728   0
3 hours ago
Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!
0 replies
ABCD1728
3 hours ago
0 replies
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Convex quadrilateral divided
gobathegreat   2
N Apr 13, 2022 by Steve12345
Source: Bosnia and Herzegovina TST 2010 problem 4
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
2 replies
gobathegreat
Mar 30, 2014
Steve12345
Apr 13, 2022
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Convex quadrilateral divided
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Reveal topic tags
geometry proposed
geometry
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Source: Bosnia and Herzegovina TST 2010 problem 4
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gobathegreat
741 posts
gobathegreat
#1 h  Mar 30, 2014, 4:11 PM • 2 Y
Y by Adventure10, Mango247
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
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delegat
652 posts
delegat
#2 h  May 14, 2014, 12:57 AM • 2 Y
Y by Adventure10, Mango247
This problem is from V. Prasolov: Problems in plane and solid geometry. For those interested in solution I have attached one solution that I have extracted from Croatian 2010. booklet.
Attachments:
Bosnian TST 2010 problem 4.pdf (205kb)
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Steve12345
620 posts
Steve12345
#3 h  Apr 13, 2022, 7:48 PM
Y by
Let the diagonals intersect at $S$. Not hard to verify that $S$ must be the excircle touchpoint of all triangles $\triangle ABC,\triangle BCD,\triangle CDA,\triangle DAB$ (by Casey's theorem). Then since $S$ divides these segments in $s-a,a$ where $s$ is the semi perimeter, we must have that all sides are equal by writing out the simple equations.
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