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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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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   1
N 2 minutes ago by Filipjack
Source: 2017 Peru Southern Cone TST P2
Determine if there exists a positive integer $n$ such that $10^n - 1$ is a divisor of $11^n - 1$.
1 reply
EmersonSoriano
2 hours ago
Filipjack
2 minutes ago
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D1018 : Can you do that ?
Dattier   1
N 4 minutes ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
4 minutes ago
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A special family of subsets of {1, 2, ..., 100}.
EmersonSoriano   0
an hour 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
1 viewing
EmersonSoriano
an hour ago
0 replies
J
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Point that is orthocenter, incenter, and centroid
EmersonSoriano   0
an hour 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
an hour ago
0 replies
J
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Divisibility involving the product and sum of two numbers
EmersonSoriano   0
an hour 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
an hour ago
0 replies
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one nice!
teomihai   0
an hour 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
an hour ago
0 replies
J
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Least common multiple from $n+1$ to $n+2016$.
EmersonSoriano   0
an hour 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
an hour ago
0 replies
J
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Moving tokens from the leftmost end to the rightmost end.
EmersonSoriano   0
an hour 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
an hour ago
0 replies
J
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A point on the midline of BC.
EmersonSoriano   0
an hour 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
an hour ago
0 replies
J
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Mundane Primes
bryanguo   10
N 2 hours 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.
2 hours ago
J
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Set of 2n real numbers divided into two groups
EmersonSoriano   0
2 hours 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
2 hours ago
0 replies
J
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intersting system with integer positive numbers
teomihai   2
N 2 hours 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
2 hours ago
teomihai
2 hours ago
J
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Non-overlapping L-tromino tiles
EmersonSoriano   0
2 hours 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
2 hours ago
0 replies
J
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Hand shaken
mathservant   0
2 hours 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
2 hours ago
0 replies
J
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J
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cos A + cos B + cos c + cos D => ABCD trapazoid or cyclic
parmenides51   1
N Apr 18, 2021 by booth
Source: 1990 British Mathematical Olympiad p3 BrMO
The angles $A, B, C, D$ of a convex quadrilateral satisfy the relation $\cos A + \cos B + \cos c + \cos D= 0$.
Prove that $ABCD$ is a trapezium or is cyclic.
1 reply
parmenides51
Apr 18, 2021
booth
Apr 18, 2021
J
cos A + cos B + cos c + cos D => ABCD trapazoid or cyclic
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Reveal topic tags
geometry
trapezoid
cyclic quadrilateral
trigonometry
BritishMathematicalOlympiad
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G H BBookmark kLocked kLocked NReply
Source: 1990 British Mathematical Olympiad p3 BrMO
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parmenides51
30629 posts
parmenides51
#1 h  Apr 18, 2021, 4:06 PM • 2 Y
Y by DAVROS, Mango247
The angles $A, B, C, D$ of a convex quadrilateral satisfy the relation $\cos A + \cos B + \cos c + \cos D= 0$.
Prove that $ABCD$ is a trapezium or is cyclic.
Z K Y
The post below has been deleted. Click to close.
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booth
471 posts
booth
#2 h  Apr 18, 2021, 5:07 PM
Y by
SInce $\cos (\frac{A+B}{2})=\cos (\frac{360-C-D}{2})=-\cos (\frac{C+D}{2})$, we have
\begin{align*} \cos A+\cos B+\cos C+\cos D&=2\cos (\frac{A+B}{2})\cos (\frac{A-B}{2})+2\cos (\frac{C+D}{2})\cos (\frac{C-D}{2})\\ &=2\cos (\frac{A+B}{2})(\cos (\frac{A-B}{2})-\cos (\frac{C-D}{2}))\\ &=2\cos (\frac{A+B}{2})\cdot (-2\sin (\frac{A-B+C-D}{2})\sin (\frac{A-B-C+D}{2}))\\ &=-4\cos (\frac{A+B}{2})\sin (\frac{A-B+C-D}{2})\sin (\frac{A-B-C+D}{2})\\ &=0 \end{align*}
This post has been edited 2 times. Last edited by booth, Apr 18, 2021, 5:09 PM
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