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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Find all such primes
Entrepreneur   2
N 4 minutes ago by straight
Source: Own
Find all primes $p,q\;\&\;r$ such that $$\color{blue}{pq=r^2+r+1.}$$
2 replies
Entrepreneur
26 minutes ago
straight
4 minutes ago
J
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Inspired by giangtruong13
sqing   5
N 13 minutes ago by kokcio
Source: Own
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ 61\leq a+b+2c+d\leq \frac{265}{3}$$$$- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$$$$\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$$
5 replies
sqing
Yesterday at 2:57 AM
kokcio
13 minutes ago
J
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four points lie on a circle
pohoatza   76
N 15 minutes ago by Bonime
Source: IMO Shortlist 2006, Geometry 2, AIMO 2007, TST 1, P2
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.

Proposed by Vyacheslev Yasinskiy, Ukraine
76 replies
1 viewing
pohoatza
Jun 28, 2007
Bonime
15 minutes ago
J
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TST Junior Romania 2025
ant_   7
N 27 minutes ago by MR.1
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
7 replies
ant_
Yesterday at 5:01 PM
MR.1
27 minutes ago
J
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NEPAL TST 2025 DAY 2
Tony_stark0094   3
N 28 minutes ago by ThatApollo777
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.
3 replies
Tony_stark0094
Today at 8:40 AM
ThatApollo777
28 minutes ago
J
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Radical Condition Implies Isosceles
peace09   8
N 32 minutes ago by cubres
Source: Black MOP 2012
Prove that any triangle with
\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]is isosceles.
8 replies
peace09
Aug 10, 2023
cubres
32 minutes ago
J
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Solllllllvvve
youochange   3
N 40 minutes ago by sadat465
Source: All Russian Olympiad 2017 Day 1 grade 10 P5
Suppose n is a composite positive integer. Let $1 = a_1 < a_2 < · · · < a_k = n$ be all the divisors of $n$. It is known, that $a_1+1, . . . , a_k+1$ are all divisors for some $m $(except $1, m$). Find all such $n.$
3 replies
youochange
Jan 12, 2025
sadat465
40 minutes ago
J
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a_n >= 1/n if a_{n+1}^2 + a_{n+1} = a_n, a_1=1 , a_i>=0
parmenides51   13
N an hour ago by Safal
Source: Canadian Junior Mathematical Olympiad - CJMO 2020 p1
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
13 replies
parmenides51
Jul 15, 2020
Safal
an hour ago
J
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Number Theory Chain!
JetFire008   33
N an hour ago by Primeniyazidayi
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
33 replies
JetFire008
Apr 7, 2025
Primeniyazidayi
an hour ago
J
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Inequality with a,b,c
GeoMorocco   1
N 2 hours ago by Natrium
Source: Morocco Training
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
1 reply
GeoMorocco
Yesterday at 10:05 PM
Natrium
2 hours ago
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IMO Shortlist 2013, Number Theory #1
lyukhson   149
N 2 hours ago by SSS_123
Source: IMO Shortlist 2013, Number Theory #1
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
149 replies
lyukhson
Jul 10, 2014
SSS_123
2 hours ago
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Tangents and chord
iv999xyz   1
N 2 hours ago by aidenkim119
Given a circle with chord AB. k and l are tangents to the circle at points A and B. C and E are in different half-planes with respect to AB and lie on k, and F and D are in different half-planes with respect to AB and lie on l. Furthermore, C and F are in the same half-plane with respect to AB and AC = BD; AE = BF. CD intersects the circle at P and R and EF intersects the circle at Q and S. P and Q are in the same half-plane with respect to AB and in different half-plane with R and S. Prove that PQRS is a parallelogram if and only if AB, CD, and EF intersect at one point.
1 reply
iv999xyz
Today at 9:41 AM
aidenkim119
2 hours ago
J
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Find the area enclosed by the curve |z|^2 + |z^2 - 2i| = 16
mqoi_KOLA   2
N 3 hours ago by mqoi_KOLA
Find the area of the Argand plane enclosed by the curve $$ |z|^2 + |z^2 - 2i| = 16.$$(ans- $3 \sqrt7 \pi$)
2 replies
mqoi_KOLA
Today at 11:58 AM
mqoi_KOLA
3 hours ago
J
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Very tight inequalities
KhuongTrang   2
N 3 hours ago by SunnyEvan
Source: own
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{35a+12b+2}+\frac{1}{35b+12c+2}+\frac{1}{35c+12a+2}\ge \frac{4}{39}.}$$$$\color{black}{\frac{1}{4a+9b+6}+\frac{1}{4b+9c+6}+\frac{1}{4c+9a+6}\le \frac{2}{9}.}$$When does equality hold?
2 replies
KhuongTrang
May 17, 2024
SunnyEvan
3 hours ago
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J
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Cutting the cube
Johann Peter Dirichlet   2
N Aug 21, 2019 by NiltonCesar
Source: Brazil Math Olympiad - 2000
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?
2 replies
Johann Peter Dirichlet
Mar 4, 2006
NiltonCesar
Aug 21, 2019
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Cutting the cube
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Reveal topic tags
geometry
3D geometry
geometry proposed
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G H BBookmark kLocked kLocked NReply
Source: Brazil Math Olympiad - 2000
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Johann Peter Dirichlet
376 posts
Johann Peter Dirichlet
#1 h  Mar 4, 2006, 8:46 PM • 2 Y
Y by Adventure10, Mango247
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?
Z K Y
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rem
1434 posts
rem
#2 h  Jul 20, 2006, 8:21 PM • 2 Y
Y by Adventure10, Mango247
Is the anwser $2^{9}=512$?(I am not sure if my solution is right)
Can someone please post the solution?
Z K Y
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NiltonCesar
161 posts
NiltonCesar
#3 h  Aug 21, 2019, 1:59 AM • 1 Y
Y by Adventure10
rem wrote:
Is the anwser $2^{9}=512$?(I am not sure if my solution is right)
Can someone please post the solution?

Actually, the official answer is $96$.
This post has been edited 1 time. Last edited by NiltonCesar, Aug 21, 2019, 1:59 AM
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