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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 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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Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   17
N 10 minutes ago by Ilikeminecraft
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
17 replies
MarkBcc168
Apr 28, 2020
Ilikeminecraft
10 minutes ago
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Inequality with a,b,c
GeoMorocco   7
N 27 minutes ago by lele0305
Source: Morocco Training
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
7 replies
+1 w
GeoMorocco
Apr 11, 2025
lele0305
27 minutes ago
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Property of the divisors of k^3 - 2
Scilyse   2
N an hour ago by Assassino9931
Source: KoMaL A. 892
Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$. Show that there exists integers $a$, $b$, $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$.

Proposed by Csongor Beke and László Bence Simon, Cambridge
2 replies
Scilyse
Jan 13, 2025
Assassino9931
an hour ago
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Centroid, altitudes and medians, and concyclic points
BR1F1SZ   1
N 2 hours ago by sami1618
Source: Austria National MO Part 1 Problem 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)
1 reply
BR1F1SZ
3 hours ago
sami1618
2 hours ago
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Something nice
KhuongTrang   31
N 2 hours ago by NguyenVanHoa29
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
31 replies
KhuongTrang
Nov 1, 2023
NguyenVanHoa29
2 hours ago
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Nordic 2025 P3
anirbanbz   8
N 2 hours ago by lksb
Source: Nordic 2025
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
8 replies
anirbanbz
Mar 25, 2025
lksb
2 hours ago
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another functional inequality?
Scilyse   32
N 2 hours ago by ihategeo_1969
Source: 2023 ISL A4
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]for every $x, y \in \mathbb R_{>0}$.
32 replies
Scilyse
Jul 17, 2024
ihategeo_1969
2 hours ago
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Mount Inequality erupts in all directions!
BR1F1SZ   1
N 2 hours ago by sami1618
Source: Austria National MO Part 1 Problem 1
Let $a$, $b$ and $c$ be pairwise distinct nonnegative real numbers. Prove that
\[ (a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4. \](Karl Czakler)
1 reply
BR1F1SZ
3 hours ago
sami1618
2 hours ago
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Division involving difference of squares
BR1F1SZ   1
N 3 hours ago by grupyorum
Source: Austria National MO Part 1 Problem 4
Determine all integers $n$ that can be written in the form
\[ n = \frac{a^2 - b^2}{b}, \]where $a$ and $b$ are positive integers.

(Walther Janous)
1 reply
BR1F1SZ
3 hours ago
grupyorum
3 hours ago
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Erasing the difference of two numbers
BR1F1SZ   0
3 hours ago
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.)

For which values of $n$ is it possible to end with only one number remaining on the board?

(Michael Reitmeir)
0 replies
BR1F1SZ
3 hours ago
0 replies
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4 wise men and 100 hats. 3 must guess their numbers
NO_SQUARES   2
N 3 hours ago by NO_SQUARES
Source: 239 MO 2025 10-11 p5
There are four wise men in a row, each sees only those following him in the row, i.e. the $1$st sees the other three, the $2$nd sees the $3$rd and $4$th, and the $3$rd sees only the $4$th. The devil has $100$ hats, numbered from $1$ to $100$, he puts one hat on each wise man, and hides the extra $96$ hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
2 replies
NO_SQUARES
Yesterday at 5:44 PM
NO_SQUARES
3 hours ago
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\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}
NO_SQUARES   2
N 3 hours ago by ektorasmiliotis
Source: 239 MO 2025 8-9 p4
Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]
2 replies
NO_SQUARES
Yesterday at 5:06 PM
ektorasmiliotis
3 hours ago
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BMO 2024 SL A4
MuradSafarli   2
N 3 hours ago by GreekIdiot
A4.
Let \(a \geq b \geq c \geq 0\) be real numbers such that \(ab + bc + ca = 3\).
Prove that:
\[ 3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c \]and determine all the cases when the equality occurs.
2 replies
MuradSafarli
Apr 27, 2025
GreekIdiot
3 hours ago
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Aime type Geo
ehuseyinyigit   0
4 hours ago
Source: Turkish First Round 2024
In a scalene triangle $ABC$, let $M$ be the midpoint of side $BC$. Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$. If $(BMN)$ is tangent to $AB$ at $B$, find $AB/MA$.
0 replies
ehuseyinyigit
4 hours ago
0 replies
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Polygon with minimum internal angle 120^\circ
Kunihiko_Chikaya   1
N Apr 28, 2025 by Mathzeus1024
How many sides can have the polygon with minimum internal angle of $ 120^\circ$ by adding every $ 5^\circ$?
Sorry for my poor English.
1 reply
Kunihiko_Chikaya
Aug 9, 2007
Mathzeus1024
Apr 28, 2025
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Polygon with minimum internal angle 120^\circ
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Kunihiko_Chikaya
14514 posts
Kunihiko_Chikaya
#1 h  Aug 9, 2007, 4:27 PM • 2 Y
Y by Adventure10, Mango247
How many sides can have the polygon with minimum internal angle of $ 120^\circ$ by adding every $ 5^\circ$?
Sorry for my poor English.
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The post below has been deleted. Click to close.
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Mathzeus1024
862 posts
Mathzeus1024
#2 h  Apr 28, 2025, 2:12 PM
Y by
For any convex polygon with $N$ sides, the total internal angle sum (in degrees) computes to $180^{\circ}(N-2)$ (i). The polygon in question has internal angles whose sum is represented by the arithmetic series: $\sum_{k=1}^{N} 120^{\circ} + 5^{\circ}(k-1)$ (ii). Equating (i) with (ii) yields a quadratic in $N$:

$180^{\circ}(N-2) = \sum_{k=1}^{N} 120^{\circ} + 5^{\circ}(k-1)$;

or $180^{\circ}(N-2) = \sum_{k=1}^{N} 115^{\circ} + 5^{\circ}k$;

or $36(N-2) = \sum_{k=1}^{N} 23 + k$;

or $36(N-2) = 23N +\frac{N(N+1)}{2}$;

or $N^2-25N+144=0$;

or $(N-9)(N-16)=0$;

or $N=9, 16$.

In order for the polygon in question to be convex we require all of its interior angles $< 180^{\circ}$. At $N=9$ the maximum angle is $120^{\circ}+5^{\circ}(9) = 165^{\circ}$ (admissible), and $N=16$ yields $120^{\circ}+5^{\circ}(16) = 200^{\circ}$ (not admissible). Hence, our desired polygon has $\boxed{9}$. sides.
This post has been edited 1 time. Last edited by Mathzeus1024, Apr 29, 2025, 9:57 AM
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