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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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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
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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interesting geo config (2/3)
Royal_mhyasd   5
N 13 minutes ago by Royal_mhyasd
Source: own
Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$. Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.
5 replies
Royal_mhyasd
Yesterday at 11:36 PM
Royal_mhyasd
13 minutes ago
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interesting geometry config (3/3)
Royal_mhyasd   2
N 13 minutes ago by Royal_mhyasd
Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$. If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$, prove that the intersection of lines $CT$ and $BS$ lies on $HE$.

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
2 replies
Royal_mhyasd
Today at 7:06 AM
Royal_mhyasd
13 minutes ago
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Convex Quadrilateral with Bisector Diagonal
matinyousefi   8
N an hour ago by lpieleanu
Source: Germany TST 2017
In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.
8 replies
matinyousefi
Apr 11, 2020
lpieleanu
an hour ago
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Kids in clubs
atdaotlohbh   0
an hour ago
There are $6k-3$ kids in a class. Is it true that for all positive integers $k$ it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
0 replies
atdaotlohbh
an hour ago
0 replies
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Turbo's en route to visit each cell of the board
Lukaluce   22
N an hour ago by HamstPan38825
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
22 replies
Lukaluce
Apr 14, 2025
HamstPan38825
an hour ago
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n lamps
pohoatza   47
N an hour ago by yayyayyay
Source: IMO Shortlist 2006, Combinatorics 1, AIMO 2007, TST 2, P1
We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i = 1$ or $ i = n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on.
Initially all the lamps are off except the leftmost one which is on.

$ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off.
$ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.
47 replies
pohoatza
Jun 28, 2007
yayyayyay
an hour ago
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prove that at least one of them is divisible by some other member of the set.
Martin.s   0
an hour ago
Given \( n + 1 \) integers \( a_1, a_2, \ldots, a_{n+1} \), each less than or equal to \( 2n \), prove that at least one of them is divisible by some other member of the set.
0 replies
Martin.s
an hour ago
0 replies
J
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estimate for \( a_1 \) is the best possible
Martin.s   0
an hour ago
Let \( a_1 < a_2 < \cdots < a_n < 2n \) be positive integers such that no one of them is divisible by any other member of the sequence. Then
\[ a_1 \geq 2^k, \]where \( k \) is defined by the inequalities
\[ 3^k < 2n < 3^{k+1}. \]This estimate for \( a_1 \) is the best possible.
0 replies
Martin.s
an hour ago
0 replies
J
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best possible estimate.
Martin.s   0
an hour ago
Let \( a_1 < a_2 < \cdots < a_n < 2n \) be a sequence of positive integers. Then
\[ \max \left( (a_i, a_j) \right) > \frac{38n}{147} - c, \]where \( c \) is a constant independent of \( n \), and \( (a_i, a_j) \) denotes the greatest common divisor of \( a_i \) and \( a_j \). This is the best possible estimate.
0 replies
Martin.s
an hour ago
0 replies
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Polynomial
Fang-jh   6
N an hour ago by yofro
Source: Chinese TST 2007 1st quiz P3
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) = 1$ and $ (x + 1)[f(x)]^2 - 1$ is an odd function.
6 replies
Fang-jh
Jan 3, 2009
yofro
an hour ago
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collinear wanted, toucpoints of incircle related
parmenides51   2
N an hour ago by Tamam
Source: 2018 Thailand October Camp 1.2
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
2 replies
parmenides51
Oct 15, 2020
Tamam
an hour ago
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show that there are at least eight vertices where exactly three edges meet.
Martin.s   0
2 hours ago
If all the faces of a convex polyhedron have central symmetry, show that there are at least eight vertices where exactly three edges meet. (The cube has exactly eight such vertices.)
0 replies
Martin.s
2 hours ago
0 replies
J
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Nice problem
Martin.s   0
2 hours ago
If \(p\) is a prime and \(n \geq p\), then
\[ n! \sum_{pi+j=n} \frac{1}{p^i i! j!} \equiv 0 \pmod{p}. \]
0 replies
Martin.s
2 hours ago
0 replies
J
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2-var inequality
sqing   12
N 2 hours ago by ytChen
Source: Own
Let $ a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq 8$$$$ (a^2+b^2)(a+1)(b+1) \leq 8$$Let $ a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$
12 replies
sqing
Yesterday at 1:35 PM
ytChen
2 hours ago
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J
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3 var inquality
sqing   2
N Apr 16, 2025 by sqing
Source: Own
Let $ a,b,c> 0 $ and $ 4(a+b) +3c-ab \geq10$ . Prove that
$$a^2+b^2+c^2+kabc\geq k+3$$Where $0\leq k \leq 1. $
$$a^2+b^2+c^2+abc\geq 4$$
2 replies
sqing
Apr 16, 2025
sqing
Apr 16, 2025
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3 var inquality
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Reveal topic tags
inequalities proposed
inequalities
algebra
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Source: Own
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sqing
42555 posts
sqing
#1 h  Apr 16, 2025, 2:49 AM
Y by
Let $ a,b,c> 0 $ and $ 4(a+b) +3c-ab \geq10$ . Prove that
$$a^2+b^2+c^2+kabc\geq k+3$$Where $0\leq k \leq 1. $
$$a^2+b^2+c^2+abc\geq 4$$
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 2:53 AM
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lbh_qys
609 posts
lbh_qys
#2 h  Apr 16, 2025, 3:12 AM
Y by
sqing wrote:
Let $ a,b,c> 0 $ and $ 4(a+b) +3c-ab \geq10$ . Prove that
$$a^2+b^2+c^2+kabc\geq k+3$$Where $0\leq k \leq 1. $
$$a^2+b^2+c^2+abc\geq 4$$

If \(abc > 1\), then \(a^2 + b^2 + c^2 > 3\), and hence
\[ a^2 + b^2 + c^2 + k\,abc > k + 3. \]
If \(c > 2\), then
\[ a^2 + b^2 + c^2 + abc \geq c^2 > 4 \geq k+3. \]
If \(abc \leq 1\) and \(c \leq 2\), then
\[ a^2 + b^2 + c^2 + k(abc) - k - 3 \geq a^2 + b^2 + c^2 + abc - 4 = (a+b+2c-4)^2 + (2-c)(4a+4b+3c-ab-10) \geq 0. \]
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sqing
42555 posts
sqing
#3 h  Apr 16, 2025, 3:19 AM
Y by
Very very nice.Thank lbh_qys.
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