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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 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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Very tasteful inequality
tom-nowy   0
41 minutes ago
Let $a,b,c \in (-1,1)$. Prove that $$(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$$
0 replies
tom-nowy
41 minutes ago
0 replies
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D1025 : Can you do that?
Dattier   3
N an hour ago by Dattier
Source: les dattes à Dattier
Let $x_{n+1}=x_n^3$ and $x_0=3$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
3 replies
Dattier
Yesterday at 8:24 PM
Dattier
an hour ago
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Parallel condition and isogonal
ItzsleepyXD   1
N an hour ago by moony_
Source: Own , Mock Thailand Mathematic Olympiad P5
Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .
1 reply
ItzsleepyXD
2 hours ago
moony_
an hour ago
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RMM 2013 Problem 1
dr_Civot   31
N an hour ago by cursed_tangent1434
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
31 replies
dr_Civot
Mar 2, 2013
cursed_tangent1434
an hour ago
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Inspired by old results
sqing   0
an hour ago
Source: Own
Let $ a , b , c>0 $and $ abc=1 $. Prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3 \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$h
0 replies
sqing
an hour ago
0 replies
J
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amazing balkan combi
egxa   7
N an hour ago by Assassino9931
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
7 replies
2 viewing
egxa
Apr 27, 2025
Assassino9931
an hour ago
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Question on Balkan SL
Fmimch   2
N an hour ago by Assassino9931
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
2 replies
Fmimch
Today at 12:13 AM
Assassino9931
an hour ago
J
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Or statement function
ItzsleepyXD   1
N an hour ago by Haris1
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
1 reply
ItzsleepyXD
2 hours ago
Haris1
an hour ago
J
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Inequlities
sqing   32
N an hour ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
32 replies
sqing
Jul 19, 2024
sqing
an hour ago
J
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Inequalities
sqing   2
N an hour ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
2 replies
sqing
Jul 12, 2024
sqing
an hour ago
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Add a digit to obtain a new perfect square
Lukaluce   2
N an hour ago by TopGbulliedU
Source: 2024 Junior Macedonian Mathematical Olympiad P4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.

Proposed by Ilija Jovčeski
2 replies
Lukaluce
Apr 14, 2025
TopGbulliedU
an hour ago
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Simple inequality
sqing   7
N 2 hours ago by sqing
Source: Daniel Sitaru
Let $a,b,c>0$ . Prove that$$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$$
7 replies
sqing
Feb 10, 2017
sqing
2 hours ago
J
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Vector Vortex
steven_zhang123   1
N 2 hours ago by Mathzeus1024
Source: NS Issue 1 P3 (2014.4)
Let $v_{1}, v_{2}, \cdots, v_{n}$ be $n$ unit vectors on a plane, where $n$ is an odd number. Prove that there exist $\varepsilon _i\in \left \{ -1,1 \right \} $ for $i=1,2,\cdots,n$ such that $\left | \sum_{i=1}^{n} \varepsilon_i v_i \right | \le 1.$
1 reply
steven_zhang123
Feb 15, 2025
Mathzeus1024
2 hours ago
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Basic geometry
AlexCenteno2007   5
N 4 hours ago by mathafou
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
5 replies
AlexCenteno2007
Feb 9, 2025
mathafou
4 hours ago
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fractional part
Ecrin_eren   3
N Apr 15, 2025 by rchokler
{x^2}+{x}=0.64

How many positive real values of x satisfy this equation?
3 replies
Ecrin_eren
Apr 13, 2025
rchokler
Apr 15, 2025
J
fractional part
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Reveal topic tags
algebra
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Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 13, 2025, 2:51 PM
Y by
{x^2}+{x}=0.64

How many positive real values of x satisfy this equation?
Z K Y
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RPCX
3 posts
RPCX
#4 h  Apr 14, 2025, 11:59 AM
Y by
Infinite,all +ve integers of the form n+0.4433981132056604 where n belong to N
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Ecrin_eren
56 posts
Ecrin_eren
#5 h  Apr 14, 2025, 2:03 PM
Y by
How did you get it
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rchokler
2972 posts
rchokler
#6 h  Apr 15, 2025, 9:26 PM
Y by
Actually, $\{x^2\}\neq\{x\}^2$ in general.

Let $f(x)=\{x^2\}+\{x\}$. For $n\geq 0$ and $0\leq k\leq 2n$, define the sets $S_{nk}=\left[\sqrt{n^2+k},\sqrt{n^2+k+1}\right)$ and the image sets $T_{nk}=f(S_{nk})$. Note that $f:S_{nk}\to T_{nk}$ is bijective.

Then $\{x\}=x-n$ and $\{x^2\}=x^2-n^2-k$.

Letting $x=n+y$ gives $x^2=n^2+2ny+y^2$ and so $\{x\}=y$ and $\{x^2\}=2ny+y^2-k$ and $T_{nk}=\left[\sqrt{n^2+k}-n,1+\sqrt{n^2+k+1}-n\right)$

The upper bound of $T_n$ is $1+\sqrt{n^2+k+1}-n>1$.

Lower bound of $T_{nk}$:
$\sqrt{n^2+k}-n\leq\frac{16}{25}\implies 25\sqrt{n^2+k}\leq 25n+16\implies 625n^2+625k\leq 625n^2+800n+256\implies k\leq\frac{800n+256}{625}$.

$(2n+1)y+y^2-k=\frac{16}{25}\implies 25y^2+25(2n+1)y-25k-16=0\implies y=\frac{-10n-5+\sqrt{100n^2+100n+100k+89}}{10}$.

So the overall solution set is $\left\{\left.\frac{-5+\sqrt{100n^2+100n+100k+89}}{10}\right|n\geq 0,0\leq k\leq\left\lfloor\frac{800n+256}{625}\right\rfloor\right\}$
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