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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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Geometry
MathsII-enjoy   4
N 11 minutes ago by whwlqkd
Given triangle $ABC$ inscribed in $(O)$ with $M$ being the midpoint of $BC$. The tangents at $B, C$ of $(O)$ intersect at $D$. Let $N$ be the projection of $O$ onto $AD$. On the perpendicular bisector of $BC$, take a point $K$ that is not on $(O)$ and different from M. Circle $(KBC)$ intersects $AK$ at $F$. Lines $NF$ and $AM$ intersect at $E$. Prove that $AEF$ is an isosceles triangle.
4 replies
MathsII-enjoy
May 15, 2025
whwlqkd
11 minutes ago
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order of a function greater than c*n-1
YLG_123   2
N an hour ago by SimplisticFormulas
Source: Brazil EGMO TST2 2024 #1
Let \( \mathbb{N} \) be the set of all positive integers. We say that a function \( f: \mathbb{N} \to \mathbb{N} \) is Georgian if \( f(1) = 1 \) and, for every positive integer \( n \), there exists a positive integer \( k \) such that
\[ f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}. \]If \( f \) is a Georgian function, we define, for each positive integer \( n \), \( \text{ord}(n) \) as the smallest positive integer \( m \) such that \( f^{(m)}(n) = 1 \). Determine all positive real numbers \( c \) for which there exists a Georgian function such that, for every positive integer \( n \geq 2024 \), it holds that \( \text{ord}(n) \geq cn - 1 \).
2 replies
YLG_123
Oct 12, 2024
SimplisticFormulas
an hour ago
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Problem 5
blug   1
N an hour ago by Tintarn
Source: Czech-Polish-Slovak Junior Match 2025 Problem 5
For every integer $n\geq 1$ prove that
$$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$
1 reply
blug
Monday at 4:53 PM
Tintarn
an hour ago
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1999 KJMO sum, square sum, cubic sum
RL_parkgong_0106   1
N 2 hours ago by JH_K2IMO
Source: 1999 KJMO
Three integers are given. $A$ denotes the sum of the integers, $B$ denotes the sum of the square of the integers and $C$ denotes the sum of cubes of the integers(that is, if the three integers are $x, y, z$, then $A=x+y+z$, $B=x^2+y^2+z^2$, $C=x^3+y^3+z^3$). If $9A \geq B+60$ and $C \geq 360$, find $A, B, C$.
1 reply
RL_parkgong_0106
Jun 30, 2024
JH_K2IMO
2 hours ago
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system of equations
JanHaj   4
N 2 hours ago by justaguy_69
Source: Kosovo National Olympiad 2025, Grade 7, Problem 2
Find all real numbers $a$ and $b$ that satisfy the system of equations:
$$\begin{cases} a &= \frac{2}{a+b} \\ \\ b &= \frac{2}{3a-b} \\ \end{cases}$$
4 replies
JanHaj
Nov 17, 2024
justaguy_69
2 hours ago
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IMO Shortlist 2012, Algebra 2
lyukhson   26
N 3 hours ago by ezpotd
Source: IMO Shortlist 2012, Algebra 2
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?

Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
26 replies
lyukhson
Jul 29, 2013
ezpotd
3 hours ago
J
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Inequality with x+y+z=1.
FrancoGiosefAG   3
N 3 hours ago by JARP091
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
3 replies
+1 w
FrancoGiosefAG
Yesterday at 8:36 PM
JARP091
3 hours ago
J
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Count the number of balanced colorings
TUAN2k8   4
N 3 hours ago by aidan0626
Source: A book
Given a $2n \times 2n$ grid ($n \in \mathbb{Z}^{+}$), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly $n$ black cells.Detemine the number of balanced colorings.
4 replies
TUAN2k8
4 hours ago
aidan0626
3 hours ago
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Sneaky one
Sunjee   5
N 3 hours ago by TBazar
Find minimum and maximum value of following function.
$$f(x,y)=\frac{\sqrt{x^2+y^2}+\sqrt{(x-2)^2+(y-1)^2}}{\sqrt{x^2+(y-1)^2}+\sqrt{(x-2)^2+y^2}} $$
5 replies
Sunjee
May 16, 2025
TBazar
3 hours ago
J
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Tangencies with cyclic quadrilateral
tapir1729   21
N 4 hours ago by Mathandski
Source: TSTST 2024, problem 4
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$.

Merlijn Staps
21 replies
1 viewing
tapir1729
Jun 24, 2024
Mathandski
4 hours ago
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Binary multiples of three
tapir1729   8
N 4 hours ago by Mathandski
Source: TSTST 2024, problem 5
For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$,
\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]Holden Mui
8 replies
tapir1729
Jun 24, 2024
Mathandski
4 hours ago
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IMO 2018 Problem 2
juckter   98
N 4 hours ago by ezpotd
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
98 replies
1 viewing
juckter
Jul 9, 2018
ezpotd
4 hours ago
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Element sum of k others
akasht   19
N 4 hours ago by ezpotd
Source: ISL 2022 A2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
19 replies
akasht
Jul 9, 2023
ezpotd
4 hours ago
J
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Least swaps to get any labeling of a regular 99-gon
Photaesthesia   9
N 5 hours ago by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations.

Proposed by Zhenhua Qu
9 replies
Photaesthesia
Nov 29, 2023
Blast_S1
5 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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Reveal topic tags
equation
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giangtruong13
148 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:
This post has been edited 7 times. Last edited by Tuvshuu, Apr 24, 2025, 2:43 PM
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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
148 posts
giangtruong13
#5 h  Apr 25, 2025, 8:19 AM
Y by
boom chalaka
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pooh123
63 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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