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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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incircle with center I of triangle ABC touches the side BC
orl   40
N 2 hours ago by Ilikeminecraft
Source: Vietnam TST 2003 for the 44th IMO, problem 2
Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)
40 replies
orl
Jun 26, 2005
Ilikeminecraft
2 hours ago
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Geometric inequality with Fermat point
Assassino9931   2
N 2 hours ago by Quantum-Phantom
Source: Balkan MO Shortlist 2024 G2
Let $ABC$ be an acute triangle and let $P$ be an interior point for it such that $\angle APB = \angle BPC = \angle CPA$. Prove that
$$ \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}} \leq \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma}. $$When does equality hold?
2 replies
Assassino9931
Yesterday at 10:21 PM
Quantum-Phantom
2 hours ago
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amazing balkan combi
egxa   2
N 2 hours ago by ja.
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
2 replies
1 viewing
egxa
Yesterday at 1:57 PM
ja.
2 hours ago
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connected set in grid
David-Vieta   5
N 2 hours ago by zmm
Source: China High School Mathematics Olympics 2024 A P3
Given a positive integer $n$. Consider a $3 \times n$ grid, a set $S$ of squares is called connected if for any points $A \neq B$ in $S$, there exists an integer $l \ge 2$ and $l$ squares $A=C_1,C_2,\dots ,C_l=B$ in $S$ such that $C_i$ and $C_{i+1}$ shares a common side ($i=1,2,\dots,l-1$).

Find the largest integer $K$ satisfying that however the squares are colored black or white, there always exists a connected set $S$ for which the absolute value of the difference between the number of black and white squares is at least $K$.
5 replies
David-Vieta
Sep 8, 2024
zmm
2 hours ago
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functional equation interesting
skellyrah   10
N 2 hours ago by jasperE3
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$$
10 replies
skellyrah
Apr 24, 2025
jasperE3
2 hours ago
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Line through orthocenter
juckter   14
N 3 hours ago by lpieleanu
Source: Mexico National Olympiad 2011 Problem 2
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.
14 replies
juckter
Jun 22, 2014
lpieleanu
3 hours ago
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Geometry with parallel lines.
falantrng   33
N 3 hours ago by joshualiu315
Source: RMM 2018,D1 P1
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
33 replies
falantrng
Feb 24, 2018
joshualiu315
3 hours ago
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subsets of {1,2,...,mn}
N.T.TUAN   9
N 3 hours ago by AshAuktober
Source: USA TST 2005, Problem 1
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.

Determine the maximum possible value of $m$ in terms of $n$.
9 replies
N.T.TUAN
May 14, 2007
AshAuktober
3 hours ago
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Combinatorics
P162008   0
3 hours ago
Source: Test Paper
If the expression for the nth term of the infinite sequence $1,2,2,3,3,3,4,4,4,4,5,.... \infty$ is $\left[\sqrt{\alpha n} + \frac{1}{\beta}\right]$ (Here $\left[.\right]$ denotes GIF) then

$1.$ Let $a = \alpha, b = \alpha + 1$ and $c = \alpha + \beta + 1$ then the number of numbers out of the first $1000$ natural numbers which are divisible by $a,b$ or $c$ is

$(A) 764$
$(B) 867$
$(C) 734$
$(D)$None of these

$2.$ Let $a = \alpha, b = \alpha + 1, c = \alpha + \beta + 1$ and $d = 3\beta + 1$. The number of divisors of the number $a^cb^cc^bd^b$ which are of the form $4n + 1, n \in N$ is equal to

$(A) 24$
$(B) 48$
$(C) 96$
$(D)$ None of these
0 replies
P162008
3 hours ago
0 replies
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The three lines AA', BB' and CC' meet on the line IO
WakeUp   45
N 3 hours ago by Ilikeminecraft
Source: Romanian Master Of Mathematics 2012
Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$.

(Russia) Fedor Ivlev
45 replies
WakeUp
Mar 3, 2012
Ilikeminecraft
3 hours ago
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Interesting inequalities
sqing   6
N 3 hours ago by pooh123
Source: Own
Let $ a,b\geq 0 $ and $ a+b+ab=3. $ Prove that
$$ab^2( b +1) \leq 4$$$$ab( b +1) \leq \frac{9}{4} $$$$a^2b ( a+b^2 ) \leq \frac{76}{27}$$$$a^2b( b +1 ) \leq \frac{3(69-11\sqrt{33})}{8} $$$$a^2b^2( b +1 ) \leq \frac{2(73\sqrt{73}-595)}{27} $$
6 replies
sqing
Yesterday at 3:12 AM
pooh123
3 hours ago
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Infimum of decreasing sequence b_n/n^2
a1267ab   34
N 4 hours ago by blueprimes
Source: USA Winter TST for IMO 2020, Problem 1 and TST for EGMO 2020, Problem 3, by Carl Schildkraut and Milan Haiman
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?

Carl Schildkraut and Milan Haiman
34 replies
a1267ab
Dec 16, 2019
blueprimes
4 hours ago
J
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Beautiful Number Theory
tastymath75025   33
N 4 hours ago by awesomehuman
Source: 2022 ISL N8
Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.
33 replies
tastymath75025
Jul 9, 2023
awesomehuman
4 hours ago
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Good Permutations in Modulo n
swynca   7
N 4 hours ago by MathLuis
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
7 replies
swynca
Yesterday at 2:03 PM
MathLuis
4 hours ago
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Inspired by lgx57
sqing   1
N Apr 14, 2025 by Primeniyazidayi
Source: Own
Let $ x,y $ be reals such that $x+y=3$ and $\frac{1}{x^2+y}+\frac{1}{x+y^2}=\frac{1}{2}$. Prove that
$$x^2+y^2=7 $$$$x^3+y^3=18 $$$$x^4+y^4=47$$
1 reply
sqing
Apr 14, 2025
Primeniyazidayi
Apr 14, 2025
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Inspired by lgx57
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Reveal topic tags
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sqing
41864 posts
sqing
#1 h  Apr 14, 2025, 2:43 PM
Y by
Let $ x,y $ be reals such that $x+y=3$ and $\frac{1}{x^2+y}+\frac{1}{x+y^2}=\frac{1}{2}$. Prove that
$$x^2+y^2=7 $$$$x^3+y^3=18 $$$$x^4+y^4=47$$
This post has been edited 1 time. Last edited by sqing, Apr 14, 2025, 2:45 PM
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Primeniyazidayi
83 posts
Primeniyazidayi
#2 h  Apr 14, 2025, 3:11 PM
Y by
Because $(x+y)^2 \geq 4xy$,it is clear that $xy \leq \frac{9}{4}$.
Expanding the second equation gives $$2(x^2+y^2) +6=x^3+y^3+xy(xy+1)=3(x^2-xy+y^2)+x^2y^2+xy$$or $$x^2y^2+x^2+y^2-2xy=6$$or $x^2y^2-4xy+3=0$.Factoring gives that $(xy-3)(xy-1)=0$.Because of the first inequality,we have $xy=1$ and putting to equation gives $x^2+y^2=7$.Then $x^3+y^3=3(x^2-xy+y^2)=18$ or just $x^3+y^3=18$.Finally,because $x^4+y^4=(x^2+y^2)^2-2x^2y^2$ we have $x^4+y^4=47$.Thus we are done.
This post has been edited 5 times. Last edited by Primeniyazidayi, Apr 14, 2025, 3:13 PM
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