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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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Cute inequality in equilateral triangle
Miquel-point   0
13 minutes ago
Source: Romanian IMO TST 1981, Day 3 P5
Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$.

Laurențiu Panaitopol
0 replies
+1 w
Miquel-point
13 minutes ago
0 replies
J
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f(x)+f([x])f({x})=x
Miquel-point   1
N 14 minutes ago by jasperE3
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
1 reply
Miquel-point
38 minutes ago
jasperE3
14 minutes ago
J
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Sum of complex fractions is an integer
Miquel-point   0
16 minutes ago
Source: Romanian IMO TST 1981, Day 3 P4
Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$.
Show that for every $a\in\mathbb{C}$ and $r>0$, the number
\[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\]is an integer. Interpet this result geometrically.

Octavian Stănășilă
0 replies
Miquel-point
16 minutes ago
0 replies
J
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Conditional upper bound for binomial coeff
Miquel-point   0
18 minutes ago
Source: Romanian IMO TST 1981, Day 3 P3
Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]for $t=1,2,\ldots,n-1$

Vasile Brânzănescu
0 replies
Miquel-point
18 minutes ago
0 replies
J
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perpendicularity involving ex and incenter
Erken   19
N 19 minutes ago by Primeniyazidayi
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
19 replies
Erken
Dec 24, 2008
Primeniyazidayi
19 minutes ago
J
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Partitioning a set so that sum of powers is balanced
Miquel-point   0
21 minutes ago
Source: Romanina IMO TST 1981, Day 3 P2
Show that a set $A$ consisting of $16$ consecutive non-negative integers can be partitioned in two disjoint sets $X$ and $Y$ each containing $8$ elements so that \(\sum\limits_{x\in X}x^k=\sum\limits_{y\in Y} y^k,\) for $k=1,2,3.$
0 replies
Miquel-point
21 minutes ago
0 replies
J
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Sum of max of some sequences is sum of powers
Miquel-point   0
24 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
24 minutes ago
0 replies
J
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Locus of sphere cutting three spheres along great circles
Miquel-point   0
28 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
28 minutes ago
0 replies
J
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s(n) and s(n+1) divisible by m
Miquel-point   0
31 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
31 minutes ago
0 replies
J
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One of first p terms in sequence isn't prime
Miquel-point   1
N 31 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
40 minutes ago
Filipjack
31 minutes ago
J
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max |sin x|, |sin (x+1)| > 1/3
Miquel-point   0
33 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
33 minutes ago
0 replies
J
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Colouring lattice points from 1981
Miquel-point   0
36 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
36 minutes ago
0 replies
J
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max(PA,PC) when ABCD square
Miquel-point   0
42 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
42 minutes ago
0 replies
J
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Polynomial divisible by x^2+1
Miquel-point   0
44 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
44 minutes ago
0 replies
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J
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Functional Equation
AnhQuang_67   5
N Yesterday at 7:07 AM by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
5 replies
AnhQuang_67
Apr 4, 2025
jasperE3
Yesterday at 7:07 AM
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Functional Equation
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Reveal topic tags
function
algebra
functional equation
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AnhQuang_67
54 posts
AnhQuang_67
#1 h  Apr 4, 2025, 4:50 PM
Y by
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
This post has been edited 1 time. Last edited by AnhQuang_67, Yesterday at 2:25 AM
Reason: oops my bad
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maromex
142 posts
maromex
#2 h  Apr 4, 2025, 4:59 PM • 1 Y
Y by eevee9406
Is that supposed to say $yf(x)$ instead of $y(x)$?
This post has been edited 1 time. Last edited by maromex, Apr 4, 2025, 4:59 PM
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jasperE3
11167 posts
jasperE3
#3 h  Apr 4, 2025, 6:05 PM
Y by
AnhQuang_67 wrote:
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R} $$

Let $P(x,y)$ be the assertion $f(-xy+f(2x+y))=xf(y)+xy$, which we get from substituting $x\mapsto2x$ in $2f\left(-\frac{xy}2+f(x+y)\right)=xf(y)+xy$.
$P(0,x)\Rightarrow f(f(x))=0\Rightarrow f(0)=f(f(f(x)))=0$
$P(-1,2)\Rightarrow f(2)=-1\Rightarrow f(-1)=f(f(2))=0$
$P(-1,1)\Rightarrow f(1)=-\frac12$
$P(1,-2)\Rightarrow f(-2)=1\Rightarrow\frac12=f(1)=f(f(-2))=0$
So no such function.
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GreekIdiot
167 posts
GreekIdiot
#4 h  Friday at 8:15 PM
Y by
maromex wrote:
Is that supposed to say $yf(x)$ instead of $y(x)$?

If $y(x)$ is a typo and supposed to be $yf(x)$ then $f(x)=0$ satisfies.
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AnhQuang_67
54 posts
AnhQuang_67
#5 h  Yesterday at 2:26 AM
Y by
maromex wrote:
Is that supposed to say $yf(x)$ instead of $y(x)$?
Sorry guys, my bad. This is $yf(x)$
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jasperE3
11167 posts
jasperE3
#6 h  Yesterday at 7:07 AM
Y by
Some progress here:
https://artofproblemsolving.com/community/c6h3348309p31047538
surprised that it's possible to solve the full equation
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