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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
1 viewing
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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EGMO magic square
Lukaluce   15
N 32 minutes ago by Assassino9931
Source: EGMO 2025 P6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?

Proposed by Paulius Aleknavičius, Lithuania
15 replies
+1 w
Lukaluce
Monday at 11:03 AM
Assassino9931
32 minutes ago
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Ant wanna come to A
Rohit-2006   2
N 39 minutes ago by zhaoli
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABCDEDEDEA$ is not valid.
2 replies
Rohit-2006
6 hours ago
zhaoli
39 minutes ago
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Sum of squared areas of polyhedron's faces...
Miquel-point   2
N an hour ago by buratinogigle
Source: KoMaL B. 5453
The faces of a convex polyhedron are quadrilaterals $ABCD$, $ABFE$, $CDHG$, $ADHE$ and $EFGH$ according to the diagram. The edges from points $A$ and $G$, respectively are pairwise perpendicular. Prove that \[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]where $[XYZW]$ denotes the area of quadrilateral $XYZW$.

Proposed by Géza Kós, Budapest
2 replies
Miquel-point
Monday at 5:44 PM
buratinogigle
an hour ago
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IMO 2014 Problem 4
ipaper   168
N an hour ago by Bonime
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
168 replies
ipaper
Jul 9, 2014
Bonime
an hour ago
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function Z to Z..
Jackson0423   3
N an hour ago by jasperE3
Let \( f : \mathbb{Z} \to \mathbb{Z} \) be a function satisfying
\[ f(f(x)) = x^2 - 6x + 6 \quad \text{for all} \quad x \in \mathbb{Z}. \]Given that
\[ f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5, \]find the value of
\[ f(0) + f(1) + f(2) + \cdots + f(6). \]
3 replies
Jackson0423
Monday at 2:49 PM
jasperE3
an hour ago
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Functional equation
tenplusten   9
N 2 hours ago by Jakjjdm
Source: Bulgarian NMO 2015 P4
Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold:
$i) $ $f (x+y)\ge f (x)+y $
$ii) $ $f (f (x))\le x $
9 replies
tenplusten
Apr 29, 2018
Jakjjdm
2 hours ago
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Prove DK and BC are perpendicular.
yunxiu   61
N 2 hours ago by Avron
Source: 2012 European Girls’ Mathematical Olympiad P1
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.

Netherlands (Merlijn Staps)
61 replies
yunxiu
Apr 13, 2012
Avron
2 hours ago
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Number Theory Chain!
JetFire008   56
N 2 hours ago by TestX01
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
56 replies
JetFire008
Apr 7, 2025
TestX01
2 hours ago
J
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European Mathematical Cup 2016 senior division problem 1
steppewolf   11
N 3 hours ago by MuradSafarli
Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum
$$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$(with $1 \le r \le s \le 2016$) is a composite number, but:
a) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$;
b) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $GCD(a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$?
$GCD(x, y)$ denotes the greatest common divisor of $x$, $y$.

Proposed by Matija Bucić
11 replies
steppewolf
Dec 31, 2016
MuradSafarli
3 hours ago
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Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Rushil   32
N 3 hours ago by InterLoop
Source: INMO 2005 Problem 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
32 replies
Rushil
Aug 23, 2005
InterLoop
3 hours ago
J
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IMO 2018 Problem 5
orthocentre   78
N 4 hours ago by chenghaohu
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
78 replies
orthocentre
Jul 10, 2018
chenghaohu
4 hours ago
J
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3-variable inequality with min(ab,bc,ca)>=1
mathwizard888   71
N 5 hours ago by Burak0609
Source: 2016 IMO Shortlist A1
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
Proposed by Tigran Margaryan, Armenia
71 replies
mathwizard888
Jul 19, 2017
Burak0609
5 hours ago
J
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IMO Shortlist 2011, Algebra 7
orl   23
N 5 hours ago by bin_sherlo
Source: IMO Shortlist 2011, Algebra 7
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that

\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]

Proposed by Titu Andreescu, Saudi Arabia
23 replies
orl
Jul 11, 2012
bin_sherlo
5 hours ago
J
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one cyclic formed by two cyclic
CrazyInMath   31
N 6 hours ago by juckter
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
31 replies
CrazyInMath
Apr 13, 2025
juckter
6 hours ago
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CD + DA = AB if ABCD cyclic, <ABC=60^o, BC=CD
parmenides51   2
N Jun 14, 2023 by Krishijivi
Source: Switzerland - Swiss MO 2006 p7
Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$
2 replies
parmenides51
Jul 18, 2020
Krishijivi
Jun 14, 2023
J
CD + DA = AB if ABCD cyclic, <ABC=60^o, BC=CD
G H J
Reveal topic tags
geometry
equal segments
cyclic quadrilateral
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Source: Switzerland - Swiss MO 2006 p7
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parmenides51
30630 posts
parmenides51
#1 h  Jul 18, 2020, 7:04 PM
Y by
Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$
Z K Y
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WolfusA
1900 posts
WolfusA
#2 h  Jul 19, 2020, 9:13 PM
Y by
$|BC|=|AD|$ and points $A,B,C,D$ lie in this order on one circle, so $|AC|>|CD|$. Hence there exists point $E$ on segmeny $AD$ such that $|CD|=|CE|$. Then triangle $BCE$ is equilateral and $\triangle ADC\equiv\triangle AEC$ by a-s-a: ray $AC^\rightarrow$ is bisector of angle $DAB$ and $\angle AEC=120^\circ=\angle ADC$. Thus $|AB|=|AE|+|BE|=|AD|+|CD|$.
Z K Y
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Krishijivi
99 posts
Krishijivi
#3 h  Jun 14, 2023, 4:51 AM
Y by
Join BD
Since ABCD cyclic, angle ABC+angle ADC=180°
angle ADC=180°-60°=120°
Let angle DBC=x, so angle angle BDC=x
So, angle BCD=180°-2x
therefore, angle ABD=60°-x
angle ADB=120°-x
In ∆ BCD, BY sine rule
sin(180°-2x)/BD=sin x/ CD
sin 2x/BD=sin x/ CD
Since ABCD is cyclic, angle BAC= 180°- angle BCD= 2x
In ∆ ABD, By sine rule ,
sin 2x/BD= sin( 60°-x)/AD= sin(120°-x)/AB
sin 2x/BD=sin x/ CD
SO, sin x/CD=sin (60°-x)/AD
Now, by addendo,
sin x/CD=sin (60°-x)/AD=
[sin x+ sin(60°-x)]/CD+AD= sin(120°-x)/AB
cos(x-30°)/CD+DA=sin (120°-x)/AB
Since, cos(x-30°)= sin(120°-x)
So, CD + DA = AB
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