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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
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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A special family of subsets of {1, 2, ..., 100}.
EmersonSoriano   0
a minute ago
Source: 2017 Peru Southern Cone TST P10
Miguel has a list consisting of several subsets of 10 elements from the set ${1,2,\dots,100}$. He says to Cecilia: "If you pick any subset of 10 elements from ${1,2,\dots,100}$, it will be disjoint with at least one subset from my list." What is the smallest possible number of subsets Miguel's list can have if what he tells Cecilia is true?
0 replies
EmersonSoriano
a minute ago
0 replies
J
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Point that is orthocenter, incenter, and centroid
EmersonSoriano   0
4 minutes ago
Source: 2017 Peru Southern Cone TST P9
Let $BXC$ be a triangle and $A_1, A_2, A_3$ points in the same plane such that $X$ is the orthocenter of triangle $A_1BC$, $X$ is the incenter of triangle $A_2BC$, and $X$ is the centroid of triangle $A_3BC$. If line $A_1A_3$ is parallel to $BC$, prove that $A_2$ is the midpoint of segment $A_1A_3$.
0 replies
EmersonSoriano
4 minutes ago
0 replies
J
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Divisibility involving the product and sum of two numbers
EmersonSoriano   0
6 minutes ago
Source: 2017 Peru Southern Cone TST P8
Determine the smallest positive integer $n$ for which the following statement is true: If positive integers $a$ and $b$ are such that $a + b$ is a multiple of $100$ and $ab$ is a multiple of $n$, then each of the numbers $a$ and $b$ is a multiple of $100$.

0 replies
EmersonSoriano
6 minutes ago
0 replies
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one nice!
teomihai   0
21 minutes ago
3 girls and 4 boys must be seated at a round table. In how many distinct ways can they be seated so that the 3 girls do not sit next to each other and there can be a maximum of 2 girls next to each other. (The table is round so the seats are not numbered.)
0 replies
teomihai
21 minutes ago
0 replies
J
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Least common multiple from $n+1$ to $n+2016$.
EmersonSoriano   0
23 minutes ago
Source: 2017 Peru Southern Cone TST P7
For each positive integer $n$, define
$$P_n = (n+1)(n+2)(n+3)\dots(n+2016)$$and
$$Q_n = \text{lcm}(n+1, n+2, n+3, \dots, n+2016),$$meaning $Q_n$ is the least common multiple of the numbers $n+1, n+2, n+3, \dots, n+2016$. Determine whether or not there exists a constant $C$ such that
$$ \frac{P_n}{Q_n} < C, $$for every positive integer $n$.
0 replies
EmersonSoriano
23 minutes ago
0 replies
J
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Moving tokens from the leftmost end to the rightmost end.
EmersonSoriano   0
27 minutes ago
Source: 2017 Peru Southern Cone TST P6
Let $n$ and $\ell$ be positive integers with $\ell > 7$. There are $n$ tokens placed initially in the leftmost cell of a horizontal row consisting of $\ell$ cells. A move consists of shifting any token $1$, $2$, $3$, $4$, $5$, or $6$ positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of $n$ and $\ell$.
0 replies
EmersonSoriano
27 minutes ago
0 replies
J
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A point on the midline of BC.
EmersonSoriano   0
32 minutes ago
Source: 2017 Peru Southern Cone TST P5
Let $ABC$ be an acute triangle with circumcenter $O$. Draw altitude $BQ$, with $Q$ on side $AC$. The parallel line to $OC$ passing through $Q$ intersects line $BO$ at point $X$. Prove that point $X$ and the midpoints of sides $AB$ and $AC$ are collinear.
0 replies
EmersonSoriano
32 minutes ago
0 replies
J
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Mundane Primes
bryanguo   10
N an hour ago by awesomeming327.
Source: 2023 HMIC P2
A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.
10 replies
bryanguo
Apr 25, 2023
awesomeming327.
an hour ago
J
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Set of 2n real numbers divided into two groups
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P4
Let $n$ be a fixed positive integer. Find the greatest real constant $C_n$ that has the following property: Any $2n$ real numbers, not necessarily distinct, that lie in the interval $[100,101]$, can be partitioned into two groups with sums $S_1$ and $S_2$ such that
$$1 \ge \frac{S_2}{S_1} \ge C_n.$$
0 replies
EmersonSoriano
an hour ago
0 replies
J
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intersting system with integer positive numbers
teomihai   2
N an hour ago by teomihai
Let next positiv integer numbers: $a ,b ,c, d $ .
If $a^4=b^3 $ , $c^5=d^6 $ and $ c-a=37 $ .
Find $ b-d$.
2 replies
teomihai
an hour ago
teomihai
an hour ago
J
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Non-overlapping L-tromino tiles
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P3
An $L$-tromino is a figure made up of three squares, obtained by removing one square from a $2\times 2$ board.

We have a $7\times 7$ board consisting of $112$ unit segments. A configuration of several $L$-trominoes is optimal if the $L$-trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two $L$-trominoes. Below is an optimal configuration of 5 $L$-trominoes:


[center]IMAGE[/center]

Determine the largest possible value of $n$ for which there exists an optimal configuration of L-trominoes on the $7\times 7$ board.
0 replies
EmersonSoriano
an hour ago
0 replies
J
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Hand shaken
mathservant   0
an hour ago
A total of 3300 handshakes were made at a party attended by 600 people. It was observed
that the total number of handshakes among any 300 people at the party is at least N. Find
the largest possible value for N.
0 replies
mathservant
an hour ago
0 replies
J
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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P2
Determine if there exists a positive integer $n$ such that $10^n - 1$ is a divisor of $11^n - 1$.
0 replies
EmersonSoriano
an hour ago
0 replies
J
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Diagonal of a pentagon that divides it into a triangle and a cyclic quadrilatera
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P1
We say that a diagonal of a convex pentagon is good if it divides the pentagon into a triangle and a circumscribable quadrilateral. What is the maximum number of good diagonals that a convex pentagon can have?

Clarification: A polygon is circumscribable if there is a circle tangent to each of its sides.
0 replies
EmersonSoriano
an hour ago
0 replies
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J
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line passing though incenter, bisects perimeter iff bisects area
parmenides51   1
N Aug 11, 2018 by BBBBMMMMLLLLE
Source: Mathematical Excalibur P179
Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .
1 reply
parmenides51
Aug 11, 2018
BBBBMMMMLLLLE
Aug 11, 2018
J
line passing though incenter, bisects perimeter iff bisects area
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Reveal topic tags
geometry
incenter
perimeter
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G H BBookmark kLocked kLocked NReply
Source: Mathematical Excalibur P179
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parmenides51
30629 posts
parmenides51
#1 h  Aug 11, 2018, 12:54 AM • 1 Y
Y by Adventure10
Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .
Z K Y
The post below has been deleted. Click to close.
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BBBBMMMMLLLLE
349 posts
BBBBMMMMLLLLE
#2 h  Aug 11, 2018, 4:40 AM • 2 Y
Y by Adventure10, Mango247
Without loss of generality, let the line intersect $AB$ at $P$ and $AC$ at $Q$.
Now the perimeter is divided in half iff $AP+AQ=s$. Then note that $[APQ] = [API]+[AQI] = r \left(\frac{AP}{2} + \frac{AQ}{2}\right) = rs/2 = [ABC]/2$, as desired. All our steps were reversible, so we have proven the converse also.
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