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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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tangent and tangent again
ItzsleepyXD   0
27 minutes ago
Source: holder send me this
Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of $\overarc{BC}$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX \parallel CY \parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN \parallel AM, \omega_b$ be a circle thaat passes through $B$, $X$ and tangents to $BC$ and $\omega_C$ be a circle that passes through $C,Y$ and tangents to $BC$. Prove that circle with center $K$ and radias $KP$ is tangent to circles $\omega_B,\omega_C$ and $\Gamma$.
0 replies
ItzsleepyXD
27 minutes ago
0 replies
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find angle
TBazar   0
34 minutes ago
Given $ABC$ triangle with $AC>BC$. We take $M$, $N$ point on AC, AB respectively such that $AM=BC$, $CM=BN$. $BM$, $AN$ lines intersect at point $K$. If $2\angle AKM=\angle ACB$, find $\angle ACB$
0 replies
TBazar
34 minutes ago
0 replies
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Geometry
Lukariman   10
N an hour ago by Captainscrubz
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
10 replies
Lukariman
Tuesday at 12:43 PM
Captainscrubz
an hour ago
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Simple inequality
sqing   21
N 2 hours ago by Bexultan
Source: JBMO 2011 Shortlist A3
$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$
21 replies
sqing
May 15, 2016
Bexultan
2 hours ago
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Primes p such that p and p^2+2p-8 are primes too
mhet49   44
N 2 hours ago by MITDragon
Source: Albanian National Math Olympiad 2012
Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.
44 replies
mhet49
Apr 1, 2012
MITDragon
2 hours ago
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Integer polynomial w factorials
Solilin   1
N 2 hours ago by Tkn
Source: 9th Thailand MO
Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.
1 reply
Solilin
Yesterday at 2:12 PM
Tkn
2 hours ago
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Combo NT
a_507_bc   4
N 2 hours ago by Namura
Source: Silk Road 2024 P1
Let $n$ be a positive integer and let $p, q>n$ be odd primes. Prove that the positive integers $1, 2, \ldots, n$ can be colored in $2$ colors, such that for any $x \neq y$ of the same color, $xy-1$ is not divisible by $p$ and $q$.
4 replies
a_507_bc
Oct 20, 2024
Namura
2 hours ago
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Old hard problem
ItzsleepyXD   2
N 3 hours ago by ItzsleepyXD
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
2 replies
ItzsleepyXD
Apr 25, 2025
ItzsleepyXD
3 hours ago
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Polynomial Factors
somebodyyouusedtoknow   1
N 3 hours ago by luutrongphuc
Source: San Diego Honors Math Contest 2025 Part II, Problem 2
Let $P(x)$ be a polynomial with real coefficients such that $P(x^n) \mid P(x^{n+1})$ for all $n \in \mathbb{N}$. Prove that $P(x) = cx^k$ for some real constant $c$ and $k \in \mathbb{N}$.
1 reply
somebodyyouusedtoknow
Apr 26, 2025
luutrongphuc
3 hours ago
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I need the technique
DievilOnlyM   15
N 5 hours ago by sqing
Let a,b,c be real numbers such that: $ab+7bc+ca=188$.
FInd the minimum value of: $5a^2+11b^2+5c^2$
15 replies
DievilOnlyM
May 23, 2019
sqing
5 hours ago
J
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Linear colorings mod 2^n
vincentwant   1
N 5 hours ago by vincentwant
Let $n$ be a positive integer. The ordered pairs $(x,y)$ where $x,y$ are integers in $[0,2^n)$ are each labeled with a positive integer less than or equal to $2^n$ such that every label is used exactly $2^n$ times and there exist integers $a_1,a_2,\dots,a_{2^n}$ and $b_1,b_2,\dots,b_{2^n}$ such that the following property holds: For any two lattice points $(x_1,y_1)$ and $(x_2,y_2)$ that are both labeled $t$, there exists an integer $k$ such that $x_2-x_1-ka_t$ and $y_2-y_1-kb_t$ are both divisible by $2^n$. How many such labelings exist?
1 reply
vincentwant
Apr 30, 2025
vincentwant
5 hours ago
J
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sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   1
N 5 hours ago by vincentwant
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases} \sqrt{n}&\text{ if n is a perfect square} \\ n+p&\text{ otherwise} \end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
1 reply
vincentwant
Apr 30, 2025
vincentwant
5 hours ago
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Flight between cities
USJL   5
N 5 hours ago by Photaesthesia
Source: 2025 Taiwan TST Round 1 Mock P5
A country has 2025 cites, with some pairs of cities having bidirectional flight routes between them. For any pair of the cities, the flight route between them must be operated by one of the companies $X, Y$ or $Z$. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities $A, B$ and $C$, with flight routes $A \leftrightarrow B$ and $A \leftrightarrow C$ operated by two different companies, then there must exist flight route $B \leftrightarrow C$ operated by the third company different from $A \leftrightarrow B$ and $A \leftrightarrow C$ .

Let $n_X$, $n_Y$ and $n_Z$ denote the number of flight routes operated by companies $X, Y$ and $Z$, respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of $\max(n_X, n_Y , n_Z)$.

Proposed by usjl and YaWNeeT
5 replies
USJL
Mar 8, 2025
Photaesthesia
5 hours ago
J
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A problem from Le Anh Vinh book.
minhquannguyen   0
5 hours ago
Source: LE ANH VINH, DINH HUONG BOI DUONG HOC SINH NANG KHIEU TOAN TAP 1 DAI SO
Let $n$ is a positive integer. Determine all functions $f:(1,+\infty)\to\mathbb{R}$ such that
\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]
0 replies
minhquannguyen
5 hours ago
0 replies
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Volume of right parallelepiped with minimal surface area
WakeUp   2
N Feb 2, 2011 by m.candales
Source: Romanian JBMO TST 2001
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
2 replies
WakeUp
Jan 19, 2011
m.candales
Feb 2, 2011
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Volume of right parallelepiped with minimal surface area
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Reveal topic tags
geometry
number theory proposed
number theory
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G H BBookmark kLocked kLocked NReply
Source: Romanian JBMO TST 2001
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WakeUp
1347 posts
WakeUp
#1 h  Jan 19, 2011, 5:53 PM • 2 Y
Y by Adventure10, Mango247
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
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mszew
1044 posts
mszew
#2 h  Jan 28, 2011, 7:26 PM • 2 Y
Y by Adventure10, Mango247
Volume: $V=abc\ge 1001$
half Area $A_2=\frac{A}{2}=ab+bc+ca$

WLOG: $a \le b \le c$

$a=b=10$ $c=11$, $V=1100$, $A_2=320$ Then $a \le 10$

If $a\le 3$ then $334 \le bc \le A_2$
then $a \in \{4,5,6,7,8,9,10\}$

Notice that $\frac{1001}{a} \le bc$
and $A_2=a(b+c)+bc$
Checking some cases to find $(a,b,c)=(7,12,12)$
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m.candales
186 posts
m.candales
#3 h  Feb 2, 2011, 4:12 PM • 2 Y
Y by Adventure10, Mango247
The right answer is $(8,9,14)$
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