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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
J
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Inequality
VicKmath7   16
N 19 minutes ago by Marcus_Zhang
Source: Serbia JBMO TST 2020 P3
Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.
16 replies
VicKmath7
Sep 5, 2020
Marcus_Zhang
19 minutes ago
J
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Finding signs in a nice inequality of L. Panaitopol
Miquel-point   1
N an hour ago by Quantum-Phantom
Source: Romanian IMO TST 1981, Day 4 P4
Consider $x_1,\ldots,x_n>0$. Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that
\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]
Laurențiu Panaitopol
1 reply
Miquel-point
Yesterday at 8:00 PM
Quantum-Phantom
an hour ago
J
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Cyclic system of equations
KAME06   4
N an hour ago by Rainbow1971
Source: OMEC Ecuador National Olympiad Final Round 2024 N3 P1 day 1
Find all real solutions:
$$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$
4 replies
KAME06
Feb 28, 2025
Rainbow1971
an hour ago
J
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Common tangent to diameter circles
Stuttgarden   2
N 4 hours ago by Giant_PT
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
2 replies
Stuttgarden
Mar 31, 2025
Giant_PT
4 hours ago
J
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functional equation
hanzo.ei   2
N 4 hours ago by MathLuis

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
2 replies
hanzo.ei
Yesterday at 6:08 PM
MathLuis
4 hours ago
J
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Geometry
youochange   5
N 4 hours ago by lolsamo
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
5 replies
youochange
Yesterday at 11:27 AM
lolsamo
4 hours ago
J
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Something nice
KhuongTrang   25
N 5 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
5 hours ago
J
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Two Functional Inequalities
Mathdreams   6
N 5 hours ago by Assassino9931
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
6 replies
Mathdreams
Yesterday at 1:34 PM
Assassino9931
5 hours ago
J
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Pythagorean new journey
XAN4   2
N 5 hours ago by mathprodigy2011
Source: Inspired by sarjinius
The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$, if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.
2 replies
XAN4
Yesterday at 3:41 AM
mathprodigy2011
5 hours ago
J
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sqrt(2) and sqrt(3) differ in at least 1000 digits
Stuttgarden   2
N 5 hours ago by straight
Source: Spain MO 2025 P3
We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.
2 replies
Stuttgarden
Mar 31, 2025
straight
5 hours ago
J
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combinatorics and number theory beautiful problem
Medjl   2
N 5 hours ago by mathprodigy2011
Source: Netherlands TST for BxMo 2017 problem 4
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$
2 replies
Medjl
Feb 1, 2018
mathprodigy2011
5 hours ago
J
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Squence problem
AlephG_64   1
N 5 hours ago by RagvaloD
Source: 2025 Finals Portuguese Math Olympiad P1
Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?
1 reply
AlephG_64
Saturday at 1:19 PM
RagvaloD
5 hours ago
J
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50 points in plane
pohoatza   12
N 5 hours ago by de-Kirschbaum
Source: JBMO 2007, Bulgaria, problem 3
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
12 replies
pohoatza
Jun 28, 2007
de-Kirschbaum
5 hours ago
J
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beautiful functional equation problem
Medjl   6
N 6 hours ago by Sadigly
Source: Netherlands TST for BxMO 2017 problem 2
Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$$f(p)=1$ for all prime numbers $p$.
$ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$
6 replies
Medjl
Feb 1, 2018
Sadigly
6 hours ago
J
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J
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2019 Nepal National Mathematics Olympiad
Piinfinity   3
N Apr 4, 2025 by godchunguus
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
3 replies
Piinfinity
Oct 13, 2020
godchunguus
Apr 4, 2025
J
2019 Nepal National Mathematics Olympiad
G H J
Reveal topic tags
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algebra
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Piinfinity
61 posts
Piinfinity
#1 h  Oct 13, 2020, 5:56 AM • 2 Y
Y by Mango247, Mango247
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
Z K Y
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hyay
181 posts
hyay
#2 h  Oct 13, 2020, 8:59 AM
Y by
Solution
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Tintarn
9029 posts
Tintarn
#3 h  Oct 13, 2020, 10:44 AM
Y by
Piinfinity wrote:
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
This is Problem 5 from Baltic Way 2011. See here.
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godchunguus
13 posts
godchunguus
#4 h  Apr 4, 2025, 2:39 AM
Y by
$\text{Let} \ P(x)\text{ denote } f(f(x)) = x^2 - x + 1. \\ \\ P(1) \implies f(f(1)) = 1 \\ \\ P(f(1)) \implies f(f(f(1))) = f(1)^2 - f(1) + 1 \Rightarrow f(1) = f(1)^2 - f(1) + 1 \Rightarrow f(1)^2 - 2f(1) + 1 = 0 \Rightarrow f(1) = 1 \\ \\ P(0) \implies f(f(0)) = 1 \\ \\ P(f(0)) \implies f(f(f(0))) = f(0)^2 - f(0) + 1 \Rightarrow f(0)^2 - f(0) = 0 \Rightarrow f(0) = 0 \text{ or } 1. \\ \\ \text{Recall that } f(f(0)) = 1. \text{This would imply that } f(0) = 1\text{, as we also know that } f(1) = 1. \\ \\ \therefore f(0) = 1.$
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