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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
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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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Inequality with three conditions
oVlad   2
N 3 minutes ago by Quantum-Phantom
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
2 replies
oVlad
Today at 1:48 PM
Quantum-Phantom
3 minutes ago
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GCD Functional Equation
pinetree1   61
N 25 minutes ago by ihategeo_1969
Source: USA TSTST 2019 Problem 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.

Ankan Bhattacharya
61 replies
1 viewing
pinetree1
Jun 25, 2019
ihategeo_1969
25 minutes ago
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An easy FE
oVlad   3
N 43 minutes ago by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
3 replies
oVlad
Today at 1:36 PM
jasperE3
43 minutes ago
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Interesting F.E
Jackson0423   12
N an hour ago by jasperE3
Show that there does not exist a function
\[ f : \mathbb{R}^+ \to \mathbb{R} \]satisfying the condition that for all \( x, y \in \mathbb{R}^+ \),
\[ f(x + y^2) \geq f(x) + y. \]

~Korea 2017 P7
12 replies
Jackson0423
Apr 18, 2025
jasperE3
an hour ago
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p^3 divides (a + b)^p - a^p - b^p
62861   49
N an hour ago by Ilikeminecraft
Source: USA January TST for IMO 2017, Problem 3
Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$.

Noam Elkies
49 replies
62861
Feb 23, 2017
Ilikeminecraft
an hour ago
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basically INAMO 2010/6
iStud   1
N an hour ago by Primeniyazidayi
Source: Monthly Contest KTOM April P1 Essay
Call $n$ kawaii if it satisfies $d(n)+\varphi(n)+1=n$ ($d(n)$ is the number of positive factors of $n$, while $\varphi(n)$ is the number of integers not more than $n$ that are relatively prime with $n$). Find all $n$ that is kawaii.
1 reply
iStud
2 hours ago
Primeniyazidayi
an hour ago
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3D geometry theorem
KAME06   0
an hour ago
Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$. Prove that:
$$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$$
0 replies
KAME06
an hour ago
0 replies
J
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Funny easy transcendental geo
qwerty123456asdfgzxcvb   1
N an hour ago by golue3120
Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.
1 reply
qwerty123456asdfgzxcvb
4 hours ago
golue3120
an hour ago
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domino question
kjhgyuio   0
an hour ago
........
0 replies
kjhgyuio
an hour ago
0 replies
J
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demonic monic polynomial problem
iStud   0
2 hours ago
Source: Monthly Contest KTOM April P4 Essay
(a) Let $P(x)$ be a monic polynomial so that there exists another real coefficients $Q(x)$ that satisfy
\[P(x^2-2)=P(x)Q(x)\]Determine all complex roots that are possible from $P(x)$
(b) For arbitrary polynomial $P(x)$ that satisfies (a), determine whether $P(x)$ should have real coefficients or not.
0 replies
iStud
2 hours ago
0 replies
J
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fun set problem
iStud   0
2 hours ago
Source: Monthly Contest KTOM April P2 Essay
Given a set $S$ with exactly 9 elements that is subset of $\{1,2,\dots,72\}$. Prove that there exist two subsets $A$ and $B$ that satisfy the following:
- $A$ and $B$ are non-empty subsets from $S$,
- the sum of all elements in each of $A$ and $B$ are equal, and
- $A\cap B$ is an empty subset.
0 replies
iStud
2 hours ago
0 replies
J
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two tangent circles
KPBY0507   3
N 2 hours ago by Sanjana42
Source: FKMO 2021 Problem 5
The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$.
Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.
3 replies
KPBY0507
May 8, 2021
Sanjana42
2 hours ago
J
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trolling geometry problem
iStud   0
2 hours ago
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral $ABCD$ with $BC<AD$ and $CD<AB$. Lines $BC$ and $AD$ intersect at $X$, and lines $CD$ and $AB$ intersect at $Y$. Let $E,F,G,H$ be the midpoints of sides $AB,BC,CD,DA$, respectively. Let $S$ and $T$ be points on segment $EG$ and $FH$, respectively, so that $XS$ is the angle bisector of $\angle{DXA}$ and $YT$ is the angle bisector of $\angle{DYA}$. Prove that $TS$ is parallel to $BD$ if and only if $AC$ divides $ABCD$ into two triangles with equal area.
0 replies
iStud
2 hours ago
0 replies
J
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My hardest algebra ever created (only one solve in the contest)
mshtand1   6
N 3 hours ago by mshtand1
Source: Ukraine IMO TST P9
Find all functions \( f: (0, +\infty) \to (0, +\infty) \) for which, for all \( x, y > 0 \), the following identity holds:
\[ f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x} \]
Proposed by Mykhailo Shtandenko
6 replies
mshtand1
Apr 19, 2025
mshtand1
3 hours ago
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Abusing surjectivity
Sadigly   7
N Apr 13, 2025 by Sadigly
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that

$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
7 replies
Sadigly
Apr 13, 2025
Sadigly
Apr 13, 2025
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Abusing surjectivity
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Reveal topic tags
algebra
functional equation
TST
AZE IzHO TST
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Sadigly
147 posts
Sadigly
#1 h  Apr 13, 2025, 6:54 PM
Y by
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that

$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
This post has been edited 1 time. Last edited by Sadigly, Apr 13, 2025, 7:01 PM
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MuradSafarli
86 posts
MuradSafarli
#2 h  Apr 13, 2025, 8:34 PM
Y by
why you posted secret Izho tst questions?
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Sadigly
147 posts
Sadigly
#3 h  Apr 13, 2025, 8:39 PM
Y by
MuradSafarli wrote:
why you posted secret Izho tst questions?

Secret :rotfl:

Anyways here is a hint:

Prove that $f$ is surjective, and consider $P(a;0)$ where $f(a)=0$
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MuradSafarli
86 posts
MuradSafarli
#4 h  Apr 13, 2025, 8:44 PM
Y by
Let P(x,y) be assertion of the given equation.
P(x,0) : f(f(x)) = (x+1)g(0)+f(0) (1)
If g(0)= 0, from substitutions P(a,0) and P(b,0) we get f(a) = f(b) -> a =
b. Hence, either g(0) = 0, or f is injective.
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alexheinis
10552 posts
alexheinis
#5 h  Apr 13, 2025, 8:50 PM
Y by
@sadigly: You can't prove that $f$ is surjective since $g\equiv 0, f\equiv c$ is a class of solutions.
If $g(y)\not=0$ for some $y$ then $(x+1)g(y)+f(y)$ is surjective hence so is $f$.
So next time be more careful when you state something as a fact.
It's a nice problem though.
This post has been edited 1 time. Last edited by alexheinis, Apr 13, 2025, 8:59 PM
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MuradSafarli
86 posts
MuradSafarli
#6 h  Apr 13, 2025, 8:59 PM
Y by
Solution page #1
Attachments:
This post has been edited 1 time. Last edited by MuradSafarli, Apr 13, 2025, 9:01 PM
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MuradSafarli
86 posts
MuradSafarli
#7 h  Apr 13, 2025, 9:00 PM
Y by
Solution page #2
Attachments:
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Sadigly
147 posts
Sadigly
#8 h  Apr 13, 2025, 9:08 PM
Y by
alexheinis wrote:
@sadigly: You can't prove that $f$ is surjective since $g\equiv 0, f\equiv c$ is a class of solutions.
If $g(y)\not=0$ for some $y$ then $(x+1)g(y)+f(y)$ is surjective hence so is $f$.
So next time be more careful when you state something as a fact.
It's a nice problem though.

Well there was going to be a small case you need to consider when you try to prove surjectivity, but that's it
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