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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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Balkan MO Shortlist official booklet
guptaamitu1   9
N 32 minutes ago by envision2017
These days I was trying to find the official booklet of Balkan MO Shortlist. But apparently, there's no big list of all Balkan shortlists for previous years. Through some sources, I have been able to find the official booklet for the following years. So if people have it for other years too, can they please put it on this thread, so that everything is in one place.
[list]
[*] 2021
[*] 2020
[*] 2019
[*] 2018
[*] 2017
[*] 2016
[/list]
9 replies
guptaamitu1
Jun 19, 2022
envision2017
32 minutes ago
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IMO ShortList 2003, combinatorics problem 4
darij grinberg   37
N 35 minutes ago by Maximilian113
Source: Problem 5 of the German pre-TST 2004, written in December 03
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
37 replies
darij grinberg
May 17, 2004
Maximilian113
35 minutes ago
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Geometric inequality with Fermat point
Assassino9931   5
N 42 minutes ago by sqing
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?
5 replies
Assassino9931
Apr 27, 2025
sqing
42 minutes ago
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Basic geometry
AlexCenteno2007   6
N 43 minutes ago by vanstraelen
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
6 replies
AlexCenteno2007
Feb 9, 2025
vanstraelen
43 minutes ago
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BMO 2024 SL A1
MuradSafarli   7
N an hour ago by sqing
A1.

Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[ \frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z} \]
holds for all positive real numbers \( a, b \) and \( c \).
7 replies
MuradSafarli
Apr 27, 2025
sqing
an hour ago
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Algebraic Manipulation
Darealzolt   1
N an hour ago by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
2 hours ago
Soupboy0
an hour ago
J
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BrUMO 2025 Team Round Problem 13
lpieleanu   1
N an hour ago by vanstraelen
Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
an hour ago
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BMO 2024 SL A4
MuradSafarli   1
N an hour ago by sqing
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.
1 reply
MuradSafarli
Apr 27, 2025
sqing
an hour ago
J
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A coincidence about triangles with common incenter
flower417477   1
N an hour ago by flower417477
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
1 reply
flower417477
an hour ago
flower417477
an hour ago
J
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Difficult combinatorics problem about distinct sums under shifts
CBMaster   1
N an hour ago by CBMaster
Source: Korea
Problem. Let $a_1, ..., a_n$ be the nonnegative integers in $\{0, 1, ..., m\}$ where $m=\left\lceil \frac{n^{2/3}}{4} \right\rceil $. Define $A=\{a_i+a_j+(j-i)|1\leq i<j\leq n\}$. Prove that $|A|\geq m$.

Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function $f(n)$ such that $f(n)=O(n^\alpha)$ where $\alpha>\frac{2}{3}$, and the statement is still true when $m=f(n)$?
Or, is there a function $f(n)$ such that $f(n)\geq C \cdot n^{2/3}$ where $C>\frac{1}{4}$, and the statement is still true when $m=f(n)$?.
1 reply
CBMaster
Apr 27, 2025
CBMaster
an hour ago
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Enjoy with this one
Halykov   0
2 hours ago
Source: own
Let \( ABC \) be a scalene triangle with circumcircle \( \Gamma \) and circumcircle \( O \). Denote \( M \) and \( N \) as the midpoints of \( AC \) and \( BC \), respectively. The altitude from \( A \) to \( BC \) intersects \( \Gamma \) again at \( D \). The line \( DN \) meets \( \Gamma \) a second time at \( T \), and \( AT \) intersects \( BC \) at \( X \). The perpendicular bisector of \( AC \) intersects \( AD \) at \( S \) and \( AB \) at \( Z \). Let the circumcircle of \( \triangle XMZ \) intersect \( BC \) again at \( Y \), and let the line \( ZY \) intersect \( AD \) at \( K \).
If \( H \) is the reflection of \( S \) over \( K \), prove that the intersection of \( CH \) and \( BM \) lies on the circumcircle of \( BOC \).
0 replies
Halykov
2 hours ago
0 replies
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Conjecture: Intersection of diagonals of cyclic quadrilateral lies on a fixed li
kieusuong   1
N 2 hours ago by Aaronjudgeisgoat
Let (O) be a fixed circle and let P be a point outside the circle.
From P, draw a line that intersects the circle at points A and B.
Now, from P, draw another line that intersects the circle at points N and M so that quadrilateral ANMB is cyclic (i.e., lies on the circle).

Let AM and BN intersect at point G. Let AN and BM intersect at point T.
Let PJ be the tangent from P to circle (O), and let J be the point of tangency.

Claim (Conjecture):
As the quadrilateral ANMB varies (still inscribed in the circle), the points T, G, and J always lie on a straight line.
Moreover, this line TJ is perpendicular to the fixed chord AB.

I believe this might be a new result and would appreciate any insights or proof ideas.
Attached is a diagram for reference.
1 reply
kieusuong
3 hours ago
Aaronjudgeisgoat
2 hours ago
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Inequality with condition a+b+c = ab+bc+ca (and special equality case)
DoThinh2001   69
N 2 hours ago by mihaig
Source: BMO 2019, problem 2
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.

(Edit: Proposed by sir Leonard Giugiuc, Romania)
69 replies
DoThinh2001
May 2, 2019
mihaig
2 hours ago
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Inequlities
sqing   33
N 2 hours ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
2 hours ago
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Equation
cip1703   2
N Jan 4, 2025 by cip1703
Is it true that the only natural numbers $x$ and $y$ for which the number $A=\sqrt{4x^2+2y+6}+\sqrt{4y^2+2x+11}$ is natural are $x=1$ and $y=3$?
How? can we prove this??
2 replies
cip1703
Jan 2, 2025
cip1703
Jan 4, 2025
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Equation
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number theory
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cip1703
33 posts
cip1703
#1 h  Jan 2, 2025, 5:08 PM
Y by
Is it true that the only natural numbers $x$ and $y$ for which the number $A=\sqrt{4x^2+2y+6}+\sqrt{4y^2+2x+11}$ is natural are $x=1$ and $y=3$?
How? can we prove this??
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Solar Plexsus
1388 posts
Solar Plexsus
#2 h  Jan 3, 2025, 2:46 AM
Y by
It is easy to prove the following lemma:

If $a$ and $b$ are natural numbers s.t $\sqrt{a} + \sqrt{b} \in \mathbb{N}$, then $a$ and $b$ are perfect squares.

According to this lemma

$(1) \;\; \sqrt{4x^2 + 2y + 6} + \sqrt{4y^2 + 2x + 11} \in \mathbb{N}$

implies there exists a pair of natural numbers $(a,b)$ s.t.

$(2) \;\; (4x^2 + 2y + 6, 4y^2 + 2x + 11) = (a^2,b^2)$.

Clearly by formulas (2)

$4x^2 + 2y + 6 \geq (2x + 2)^2$,

$4y^2 + 2x + 11 \geq (2y + 1)^2$,

yielding

$(4x^2 + 2y + 6) + (4y^2 + 2x + 11) \geq (2x^2 + 2)^2 + (2y^2 + 1) = 4x^2 + 8x + 4y^2 + 4y + 5$,

i.e.

$(3) \;\; y \leq 3(2 - x)$,

which means $x<2$ (since $y>0$). Hence $x=1$, which inserted in inequality (3) and formulas (2) give us $y \leq 3$ and

$10 + 2y = a^2 \geq 4^2 =16$,

i.e. $y \geq 3$, which implies $y=3$.

Conclusion: The only pair $(x,y)$ of natural numbers satisfying condition (1) is $(x,y) = (1,3)$. q.e.d.
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cip1703
33 posts
cip1703
#3 h  Jan 4, 2025, 5:58 PM
Y by
Solar Plexsus, Thank you for this wonderful solution.
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