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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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Turkish MO 1994 P5
xeroxia   10
N a minute ago by AshAuktober
Source: Turkish Mathematical Olympiad 2nd Round 1994
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]
10 replies
xeroxia
Sep 27, 2006
AshAuktober
a minute ago
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Cyclic points [variations on a Fuhrmann generalization]
shobber   24
N 7 minutes ago by ali123456
Source: China TST 2006
$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.
24 replies
shobber
Jun 18, 2006
ali123456
7 minutes ago
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source own
Bet667   1
N 16 minutes ago by Bet667
Let $x,y\ge 0$ such that $2(x+y)=1+xy$ then find minimal value of $$x+\frac{1}{x}+\frac{1}{y}+y$$
1 reply
Bet667
17 minutes ago
Bet667
16 minutes ago
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ABC is similar to XYZ
Amir Hossein   53
N 32 minutes ago by ali123456
Source: China TST 2011 - Quiz 2 - D2 - P1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
53 replies
Amir Hossein
May 20, 2011
ali123456
32 minutes ago
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Combinatorics
TUAN2k8   1
N 33 minutes ago by TUAN2k8
A sequence of integers $a_1,a_2,...,a_k$ is call $k-balanced$ if it satisfies the following properties:
$i) a_i \neq a_j$ and $a_i+a_j \neq 0$ for all indices $i \neq j$.
$ii) \sum_{i=1}^{k} a_i=0$.
Find the smallest integer $k$ for which: Every $k-balanced$ sequence, there always exist two terms whose diffence is not less than $n$. (where $n$ is given positive integer)
1 reply
TUAN2k8
Today at 8:22 AM
TUAN2k8
33 minutes ago
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Euler line of incircle touching points /Reposted/
Eagle116   3
N 39 minutes ago by Eagle116
Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC$, $CA$, $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$.
3 replies
Eagle116
2 hours ago
Eagle116
39 minutes ago
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IMO Shortlist 2012, Number Theory 1
lyukhson   44
N 41 minutes ago by g0USinsane777
Source: IMO Shortlist 2012, Number Theory 1
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.

Proposed by Warut Suksompong, Thailand
44 replies
lyukhson
Jul 29, 2013
g0USinsane777
41 minutes ago
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Easy Geometry
Itoz   2
N an hour ago by Lil_flip38
Source: Own
$\triangle ABC$ satisfies $AC>BC$, and its incenter is $I$. $D$ is the foot from $I$ to $AC$，and $E$ is a point on segment $AC$ such that $BC=CE$. Line $BD$ meets $\odot (BCE)$ at point $F(\ne B)$, and the orthocenter of $\triangle AEI$ is $H$.

Prove that $\odot(AEH)$ is tangent to $\odot(CDF)$.
2 replies
Itoz
3 hours ago
Lil_flip38
an hour ago
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Common chord bisects segment
mofumofu   11
N an hour ago by sttsmet
Source: China TSTST 3 Day 1 Q2
Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.
11 replies
mofumofu
Mar 18, 2017
sttsmet
an hour ago
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MONT pg 31 example 1.10.40
Jaxman8   1
N an hour ago by hanulyeongsam
Can somebody explain why it works.
1 reply
Jaxman8
Thursday at 9:33 PM
hanulyeongsam
an hour ago
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Interesting F.E
Jackson0423   4
N an hour ago by Jackson0423
Show that there does not exist a function
\[ f : \mathbb{R}^+ \to \mathbb{R} \]satisfying the condition that for all \( x, y \in \mathbb{R}^+ \),
\[ f(x^2 + y) \geq f(x) + y. \]

~Korea 2017 P7
4 replies
Jackson0423
Yesterday at 4:12 PM
Jackson0423
an hour ago
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Reflections of AB, AC with respect to BC and angle bisector of A
falantrng   27
N an hour ago by ravengsd
Source: BMO 2024 Problem 1
Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the
$A$-angle bisector on $BC$. The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points
$E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ at $G$ and $H$ respectively such that $G$
lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$. Prove that the circumcircles of
$\triangle EDG$ and $\triangle FDH$ are tangent to each other.
27 replies
falantrng
Apr 29, 2024
ravengsd
an hour ago
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Test from Côte d'Ivoire Diophantine equation
MENELAUSS   0
an hour ago
determine all triplets $(x;y;z)$ of natural numbers such that
$$y \quad \text{is prime }$$
$$y \quad \text{and} \quad 3 \quad \text{does not divide} \quad z$$
$$x^3-y^3=z^2$$
0 replies
MENELAUSS
an hour ago
0 replies
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Substitutions inequality?
giangtruong13   2
N an hour ago by giangtruong13
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
2 replies
giangtruong13
Yesterday at 2:07 PM
giangtruong13
an hour ago
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2019 Polynomial problem
srnjbr   2
N Mar 30, 2025 by srnjbr
suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
2 replies
srnjbr
Mar 25, 2025
srnjbr
Mar 30, 2025
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2019 Polynomial problem
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Reveal topic tags
polynomial
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srnjbr
61 posts
srnjbr
#1 h  Mar 25, 2025, 6:15 PM
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suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
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pco
23503 posts
pco
#2 h  Mar 27, 2025, 11:19 AM
Y by
srnjbr wrote:
suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
Preliminary claim : $\forall A\in[0,t^n)$ ($n$ positive integer), $\exists Q(x)$ polynomial with coefficients $0,1$ and degree $<n$ such that $A-Q(t)\in[0,t)$
Proof

Let then $n$ such that $\sum_{i=0}^nt^i\ge 2018>\sum_{i=0}^{n-1}t^i$
Let $A=\frac{\sum_{i=0}^nt^i-2018}2$ : $A\in[0,\frac{t^n}2)\subset [0,t^{n-1})$
And so, using claim, $\exists Q(x)$ polynomial with coefficients $0,1$ and degree $<n-1$ such that $A-Q(t)\in[0,t)$
So $2A-2Q(t)\in[0,2t)$
Which is $\sum_{i=0}^nt^i-2018-2Q(t)\in[0,2t)$
$\implies$ $\sum_{i=0}^nt^i-2Q(t)\in[2018,2018+2t)\subset[2018,2020]$
$\implies$ $\sum_{i=0}^nt^i-2Q(t)-2019\in[-1,1]$
And $P(x)=\sum_{i=0}^nx^i-2Q(x)$ is a polynomial whose all coefficients are $\pm 1$

Q.E.D
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srnjbr
61 posts
srnjbr
#3 h  Mar 30, 2025, 12:58 PM
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THANK YOU
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