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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
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
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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2 var inquality
Iveela   19
N 5 minutes ago by sqing
Source: Izho 2025 P1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
19 replies
Iveela
Jan 14, 2025
sqing
5 minutes ago
J
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Set of perfect powers is irreducible
Assassino9931   0
6 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P4
For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers good if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good.

(In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.)

Lajos Hajdu and Andras Sarkozy, Hungary
0 replies
Assassino9931
6 minutes ago
0 replies
J
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Al-Khwarizmi birth year in a combi process
Assassino9931   0
8 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P3
On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad can take all the candies from $9$ consecutive non-empty baskets, while Sevinch can take all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions.

Shubin Yakov, Russia
0 replies
Assassino9931
8 minutes ago
0 replies
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Grouping angles in a pentagon with bisectors
Assassino9931   0
10 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$.

Miroslav Marinov, Bulgaria
0 replies
Assassino9931
10 minutes ago
0 replies
J
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Nice Functional Equations (ISI 2021)
integrated_JRC   11
N 12 minutes ago by lakshya2009
Source: ISI 2021 P2
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations (A) and (B) below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*}for all integers $x,y$.

(i) Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.
11 replies
integrated_JRC
Jul 18, 2021
lakshya2009
12 minutes ago
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Anything real in this system must be integer
Assassino9931   0
12 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
0 replies
Assassino9931
12 minutes ago
0 replies
J
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Tangent circles
Sadigly   1
N 19 minutes ago by lbh_qys
Source: Azerbaijan Junior MO 2025 P6
Let $T$ be a point outside circle $\omega$ centered at $O$. Tangents from $T$ to $\omega$ touch $\omega$ at $A;B$. Line $TO$ intersects bigger $AB$ arc at $C$.The line drawn from $T$ parallel to $AC$ intersects $CB$ at $E$. Ray $TE$ intersects small $BC$ arc at $F$. Prove that the circumcircle of $OEF$ is tangent to $\omega$.
1 reply
1 viewing
Sadigly
2 hours ago
lbh_qys
19 minutes ago
J
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Number theory easy
Doc.AK   10
N 26 minutes ago by MR.1
Source: JBMO shortlist 2015
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$
10 replies
Doc.AK
May 22, 2016
MR.1
26 minutes ago
J
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Inspired by Azerbaijan 2025
sqing   0
an hour ago
Source: Own
Let $ 0\leq x,y,z\leq 2. $ Prove that
$$-32\leq (x-yz)(2y-zx)(z-xy)\leq 4$$$$-64\leq (2x-yz)(y-zx)(2z-xy)\leq 24-16\sqrt 2$$$$-96\leq (2x-yz)(3y-zx)(2z-xy)\leq\frac{9}{4}$$



0 replies
sqing
an hour ago
0 replies
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Inequality
lgx57   0
an hour ago
Source: Own
$a,b>0$,$a^4+a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.

$a,b>0$,$a^4-a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.
0 replies
lgx57
an hour ago
0 replies
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Inequality
Sadigly   2
N an hour ago by sqing
Source: Azerbaijan Junior MO 2025 P5
For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire:


$$(2x-yz)(2y-zx)(2z-xy)$$
2 replies
Sadigly
2 hours ago
sqing
an hour ago
J
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Inspired by lgx57
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b>0, a^4+ab+b^4=10 $. Prove that
$$ \sqrt{10}\leq a^2+ab+b^2 \leq 6$$$$ 2\leq a^2-ab+b^2 \leq \sqrt{10}$$$$ 4\sqrt{10}\leq 4a^2+ab+4b^2 \leq18$$$$ 12<4a^2-ab+4b^2 \leq14$$
3 replies
sqing
Yesterday at 2:19 PM
sqing
an hour ago
J
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Interesting functional equation with geometry
User21837561   1
N an hour ago by User21837561
Source: BMOSL 2025 G7
For an acute triangle $ABC$, let $O$ be the circumcentre, $H$ be the orthocentre, and $G$ be the centroid.
Let $f:\pi\rightarrow\mathbb R$ satisfy the following condition:
$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$
Prove that $f$ is constant.
1 reply
User21837561
an hour ago
User21837561
an hour ago
J
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Functional Inequality Implies Uniform Sign
peace09   32
N 2 hours ago by lelouchvigeo
Source: 2023 ISL A2
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.

Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
32 replies
peace09
Jul 17, 2024
lelouchvigeo
2 hours ago
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J
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Polish MO 2007 Second Round Day 1, problem 2
robpal   2
N Mar 19, 2018 by WolfusA
$ABCDE$ is a convex pentagon and:
\[BC=CD, \;\;\; DE=EA, \;\;\; \angle BCD=\angle DEA=90^{\circ}\]
Prove, that it is possible to build a triangle from segments $AC$, $CE$, $EB$. Find the value of its angles if $\angle ACE=\alpha$ and $\angle BEC=\beta$.
2 replies
robpal
Feb 28, 2007
WolfusA
Mar 19, 2018
J
Polish MO 2007 Second Round Day 1, problem 2
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robpal
37 posts
robpal
#1 h  Feb 28, 2007, 7:59 PM • 2 Y
Y by Adventure10, Mango247
$ABCDE$ is a convex pentagon and:
\[BC=CD, \;\;\; DE=EA, \;\;\; \angle BCD=\angle DEA=90^{\circ}\]
Prove, that it is possible to build a triangle from segments $AC$, $CE$, $EB$. Find the value of its angles if $\angle ACE=\alpha$ and $\angle BEC=\beta$.
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dondigo
684 posts
dondigo
#2 h  Mar 14, 2007, 5:57 PM • 3 Y
Y by Adventure10, Adventure10, Mango247
http://www.mathlinks.ro/Forum/viewtopic.php?t=136488
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WolfusA
1900 posts
WolfusA
#3 h  Mar 19, 2018, 3:08 PM • 2 Y
Y by Adventure10, Mango247
The first part is possible to prove using trigonometry: set $BC=CD=a\wedge DE=EA=b\wedge \angle BDA=\gamma$. It's possible to write equations for $AC$, $CE$, $EB$ using law of cosines and only mentioned symbols. Now you just have to prove that triangle triangle inequality for segments of these lengths.
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