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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
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
Thursday at 11:16 PM
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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JBMO Shortlist 2022 A2
Lukaluce   13
N 24 minutes ago by Rayvhs
Source: JBMO Shortlist 2022
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by Petar Filipovski, Macedonia
13 replies
Lukaluce
Jun 26, 2023
Rayvhs
24 minutes ago
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A very beautiful geo problem
TheMathBob   4
N an hour ago by ravengsd
Source: Polish MO Finals P2 2023
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$Prove that $YI$ is the angle bisector of $XYA$.
4 replies
TheMathBob
Mar 29, 2023
ravengsd
an hour ago
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Inspired by old results
sqing   6
N an hour ago by Jamalll
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
6 replies
sqing
Apr 29, 2025
Jamalll
an hour ago
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A Duality Operation on Decreasing Integer Sequences
Ritangshu   0
an hour ago
Let \( S \) be the set of all sequences \( (a_1, a_2, \ldots) \) of non-negative integers such that
(i) \( a_1 \geq a_2 \geq \cdots \); and
(ii) there exists a positive integer \( N \) such that \( a_n = 0 \) for all \( n \geq N \).

Define the dual of the sequence \( (a_1, a_2, \ldots) \in S \) to be the sequence \( (b_1, b_2, \ldots) \), where, for \( m \geq 1 \),
\( b_m \) is the number of \( a_n \)'s which are greater than or equal to \( m \).

(i) Show that the dual of a sequence in \( S \) belongs to \( S \).

(ii) Show that the dual of the dual of a sequence in \( S \) is the original sequence itself.

(iii) Show that the duals of distinct sequences in \( S \) are distinct.
0 replies
Ritangshu
an hour ago
0 replies
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Property of a function
Ritangshu   0
an hour ago
Let \( f(x, y) = xy \), where \( x \geq 0 \) and \( y \geq 0 \).
Prove that the function \( f \) satisfies the following property:

\[ f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\} \]
for all \( (x, y) \ne (x', y') \) and for all \( \lambda \in (0, 1) \).

0 replies
Ritangshu
an hour ago
0 replies
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Subset of digits to express as a sum
anantmudgal09   46
N an hour ago by anudeep
Source: INMO 2020 P3
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.

Proposed by Sutanay Bhattacharya

Original Wording
46 replies
anantmudgal09
Jan 19, 2020
anudeep
an hour ago
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A problem on functions on sets
Ritangshu   0
an hour ago
For a finite set $A$, let $|A|$ denote the number of elements in the set $A$.

(a) Let $F$ be the set of all functions
\[ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, k\} \quad \text{with } n \geq 3,\; k \geq 2 \]satisfying the condition:
\[ f(i) \ne f(i+1) \quad \text{for every } i,\; 1 \leq i \leq n-1. \]Show that
\[ |F| = k(k-1)^{n-1}. \]
(b) Let $c(n, k)$ denote the number of functions in $F$ satisfying $f(n) \ne f(1)$.
For $n \geq 4$, show that
\[ c(n, k) = k(k-1)^{n-1} - c(n-1, k). \]
(c) Using part (b), prove that for $n \geq 3$,
\[ c(n, k) = (k-1)^n - (-1)^n(k-1). \]
0 replies
Ritangshu
an hour ago
0 replies
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Arbitrary point on BC and its relation with orthocenter
falantrng   29
N an hour ago by optimusprime154
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
29 replies
falantrng
Apr 27, 2025
optimusprime154
an hour ago
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Strange angle condition and concyclic points
lminsl   127
N an hour ago by reni_wee
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
127 replies
lminsl
Jul 16, 2019
reni_wee
an hour ago
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IMO Shortlist 2009 - Problem C5
April   37
N 2 hours ago by ihategeo_1969
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?

Proposed by Gerhard Woeginger, Netherlands
37 replies
April
Jul 5, 2010
ihategeo_1969
2 hours ago
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Divisibility of a triple
goodar2006   53
N 2 hours ago by cursed_tangent1434
Source: Iran TST 2013-First exam-2nd day-P5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?

Proposed by Mahan Malihi
53 replies
goodar2006
Apr 19, 2013
cursed_tangent1434
2 hours ago
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Sequences problem
BBNoDollar   0
2 hours ago
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
2 hours ago
0 replies
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   33
N 2 hours ago by ioannism45
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
33 replies
falantrng
Apr 27, 2025
ioannism45
2 hours ago
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Solution needed ASAP
UglyScientist   8
N 2 hours ago by deltapc
$ABC$ is acute triangle. $H$ is orthocenter, $M$ is the midpoint of $BC$, $L$ is the midpoint of smaller arc $BC$. Point $K$ is on $AH$ such that, $MK$ is perpendicular to $AL$. Prove that: $HMLK$ is paralelogram(Synthetic sol needed).
8 replies
UglyScientist
Today at 1:18 PM
deltapc
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
140 posts
giangtruong13
#1 h  Apr 24, 2025, 1:44 PM
Y by
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:
This post has been edited 7 times. Last edited by Tuvshuu, Apr 24, 2025, 2:43 PM
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navier3072
112 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
140 posts
giangtruong13
#5 h  Apr 25, 2025, 8:19 AM
Y by
boom chalaka
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pooh123
37 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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