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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
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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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Sum of max of some sequences is sum of powers
Miquel-point   0
2 minutes ago
Source: Romanian IMO TST 1981, Day 3 P1
Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$. Show that
\[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\]
Ioan Tomescu
0 replies
Miquel-point
2 minutes ago
0 replies
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Locus of sphere cutting three spheres along great circles
Miquel-point   0
6 minutes ago
Source: Romanian IMO TST 1981, Day 2 P3
Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$.

Stere Ianuș
0 replies
Miquel-point
6 minutes ago
0 replies
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s(n) and s(n+1) divisible by m
Miquel-point   0
9 minutes ago
Source: Romanian IMO TST 1981, Day 2 P2
Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$, where $s(x)$ is the sum of digits of $x$.

Dorel Miheț
0 replies
Miquel-point
9 minutes ago
0 replies
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One of first p terms in sequence isn't prime
Miquel-point   1
N 9 minutes ago by Filipjack
Source: Romanian IMO TST 1981, Day 1 P3
Let $p>2$ be a prime number, and $(a_k)_{k\geqslant 1}$ be a sequence defined by $a_1=p$ and $a_{k+1}=2a_k+1$, $k\geqslant 1$. Show that one of the first $p$ terms of the sequence is not prime.

Marcel Țena
1 reply
Miquel-point
19 minutes ago
Filipjack
9 minutes ago
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max |sin x|, |sin (x+1)| > 1/3
Miquel-point   0
12 minutes ago
Source: Romanian IMO TST 1981, Day 2 P1
Show that for every real number $x$ we have
\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]
0 replies
Miquel-point
12 minutes ago
0 replies
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Colouring lattice points from 1981
Miquel-point   0
14 minutes ago
Source: Romanian IMO TST 1981, Day 1 P5
Consider the set $S$ of lattice points with positive coordinates in the plane. For each point $P(a,b)$ from $S$, we draw a segment between it and each of the points in the set \[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]Show that there is no colouring of the points in $S$ with a finite number of colours such that every two points joined by a segment are coloured with different colours.

Ioan Tomescu
0 replies
Miquel-point
14 minutes ago
0 replies
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f(x)+f([x])f({x})=x
Miquel-point   0
16 minutes ago
Source: Romanian IMO TST 1981, Day 1 P4
Determine the function $f:\mathbb{R}\to\mathbb{R}$ such that $\forall x\in\mathbb{R}$ \[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\]and draw its graph. Find all $k\in\mathbb{R}$ for which the equation $f(x)+mx+k=0$ has solutions for any $m\in\mathbb{R}$.

V. Preda and P. Hamburg
0 replies
1 viewing
Miquel-point
16 minutes ago
0 replies
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max(PA,PC) when ABCD square
Miquel-point   0
20 minutes ago
Source: Romanian IMO TST 1981, P2 Day 1
Determine the set of points $P$ in the plane of a square $ABCD$ for which \[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]
Titu Andreescu and I.V. Maftei
0 replies
Miquel-point
20 minutes ago
0 replies
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Polynomial divisible by x^2+1
Miquel-point   0
22 minutes ago
Source: Romanian IMO TST 1981, P1 Day 1
Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]is divisible by $X^2+1$.

Mircea Becheanu
0 replies
Miquel-point
22 minutes ago
0 replies
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functional equation
hanzo.ei   0
27 minutes ago

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
0 replies
1 viewing
hanzo.ei
27 minutes ago
0 replies
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Sum of complex numbers over plus/minus
Miquel-point   0
28 minutes ago
Source: RNMO 1980 10.2
Show that if $z_1,z_2,z_3\in\mathbb C$ then
\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]Generalize the problem.

0 replies
Miquel-point
28 minutes ago
0 replies
J
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Locus problem with circles in space
Miquel-point   0
31 minutes ago
Source: RNMO 1979 10.4
Consider two circles $\mathcal C_1$ and $\mathcal C_2$ lying in parallel planes. Describe the locus of the midpoint of $M_1M_2$ when $M_i$ varies along $\mathcal C_i$ for $i=1,2$.

Ioan Tomescu
0 replies
Miquel-point
31 minutes ago
0 replies
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Functional equation
Nima Ahmadi Pour   97
N 31 minutes ago by EpicBird08
Source: ISl 2005, A2, Iran prepration exam
We denote by $\mathbb{R}^+$ the set of all positive real numbers.

Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:
\[f(x)f(y)=2f(x+yf(x))\]
for all positive real numbers $x$ and $y$.

Proposed by Nikolai Nikolov, Bulgaria
97 replies
Nima Ahmadi Pour
Apr 24, 2006
EpicBird08
31 minutes ago
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Very Cute Functional Equation :)
YLG_123   2
N an hour ago by bin_sherlo
Source: Olimphíada 2021 - Problem 6
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$:
$$f(mf(n)) + f(n) | mn + f(f(n)).$$
2 replies
YLG_123
Jul 9, 2023
bin_sherlo
an hour ago
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cut any triangle into 5 congruent triangles and a square
parmenides51   0
Jun 13, 2021
Source: 2018 1st Kharkiv Masters Tournament VII 3.2 Ukraine
Cut any triangle into $5$ congruent triangles and a square.
0 replies
parmenides51
Jun 13, 2021
0 replies
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cut any triangle into 5 congruent triangles and a square
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Reveal topic tags
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congruent triangles
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Kharkiv Masters
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Source: 2018 1st Kharkiv Masters Tournament VII 3.2 Ukraine
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parmenides51
30629 posts
parmenides51
#1 h  Jun 13, 2021, 8:23 PM
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Cut any triangle into $5$ congruent triangles and a square.
This post has been edited 4 times. Last edited by parmenides51, Dec 28, 2022, 9:35 PM
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