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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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Mundane Primes
bryanguo   10
N 13 minutes 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.
13 minutes ago
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Set of 2n real numbers divided into two groups
EmersonSoriano   0
25 minutes 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
25 minutes ago
0 replies
J
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intersting system with integer positive numbers
teomihai   2
N 32 minutes 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
32 minutes ago
J
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Non-overlapping L-tromino tiles
EmersonSoriano   0
36 minutes 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
36 minutes ago
0 replies
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Hand shaken
mathservant   0
39 minutes 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
39 minutes ago
0 replies
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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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2025 Caucasus MO Juniors P4
BR1F1SZ   1
N an hour ago by sami1618
Source: Caucasus MO
In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal, and they intersect at $E$. Perpendicular bisectors of $AB$ and $CD$ intersect at point $P$ lying inside triangle $AED$, and perpendicular bisectors of $BC$ and $DA$ intersect at point $Q$ lying inside triangle $CED$. Prove that $\angle PEQ = 90^\circ$.
1 reply
BR1F1SZ
Mar 26, 2025
sami1618
an hour ago
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functional equations over positive rationals make me big sad
bryanguo   8
N an hour ago by awesomeming327.
Source: 2023 HMIC P1
Let $\mathbb{Q}^{+}$ denote the set of positive rational numbers. Find, with proof, all functions $f:\mathbb{Q}^+ \to \mathbb{Q}^+$ such that, for all positive rational numbers $x$ and $y,$ we have \[f(x)=f(x+y)+f(x+x^2f(y)).\]
8 replies
bryanguo
Apr 25, 2023
awesomeming327.
an hour ago
J
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Two midpoints and the circumcenter are collinear.
ricarlos   0
an hour ago
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point on the perpendicular bisector of $AB$ (see figure) and $Q$, $R$ be the intersections of the perpendicular bisectors of $AC$ and $BC$, respectively, with $PA$ and $PB$. Prove that the midpoints of $PC$ and $QR$ and the point $O$ are collinear.

0 replies
ricarlos
an hour ago
0 replies
J
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Valuable subsets of segments in [1;n]
NO_SQUARES   1
N 2 hours ago by NO_SQUARES
Source: Russian May TST to IMO 2023; group of candidates P6; group of non-candidates P8
The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
1 reply
NO_SQUARES
Thursday at 8:34 PM
NO_SQUARES
2 hours ago
J
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geometry party
pnf   0
2 hours ago
https://artofproblemsolving.com/community/c4260013_geometry_party
0 replies
pnf
2 hours ago
0 replies
J
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All heads to tails?
smartvong   0
2 hours ago
Source: CEMC Euclid Contest 2025
An equilateral triangle is formed using $n$ rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to $n$ coins in the $n$th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown.

IMAGE

Below (a), (b) and (c), you will find instructions about how to refer to these turns in your solutions.

(a) If there are 3 rows of coins, give a sequence of 4 turns that results in a win.

(b) Suppose that there are 4 rows of coins. Determine whether or not there is a sequence of turns that results in a win.

(c) Determine all values of $n$ for which it is possible to win the game starting with $n$ rows of coins.

Note: For a triangle with 4 rows of coins, there are 9 possibilities for the set of three coins that Carley can flip on a given turn. These 9 possibilities are shown as shaded triangles below:

IMAGE

IMAGE

[You should use the names for these moves shown inside the 9 shaded triangles when answering (b). You should adapt this naming convention in a suitable way when answering parts (a) and (c).]
0 replies
smartvong
2 hours ago
0 replies
J
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Inequality with three variables
crazyfehmy   14
N 2 hours ago by TopGbulliedU
Source: Turkey JBMO Team Selection Test 2013, P4
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that

\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
14 replies
crazyfehmy
May 31, 2013
TopGbulliedU
2 hours ago
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J
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triangle ABC wanted, AB + AC = 2, AD + BC = \sqrt5, AD altitude
parmenides51   1
N Apr 22, 2021 by zbghj2
Source: 1979 British Mathematical Olympiad p1 BrMO
Find all triangles $ABC$ for which $AB + AC = 2$ cm. and $AD + BC = \sqrt5$ cm. , where $AD$ is the altitude through $A$, meeting $BC$ at right angles in $D$.
1 reply
parmenides51
Apr 19, 2021
zbghj2
Apr 22, 2021
J
triangle ABC wanted, AB + AC = 2, AD + BC = \sqrt5, AD altitude
G H J
Reveal topic tags
geometry
BritishMathematicalOlympiad
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G H BBookmark kLocked kLocked NReply
Source: 1979 British Mathematical Olympiad p1 BrMO
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parmenides51
30629 posts
parmenides51
#1 h  Apr 19, 2021, 7:36 PM • 1 Y
Y by samrocksnature
Find all triangles $ABC$ for which $AB + AC = 2$ cm. and $AD + BC = \sqrt5$ cm. , where $AD$ is the altitude through $A$, meeting $BC$ at right angles in $D$.
Z K Y
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zbghj2
89 posts
zbghj2
#2 h  Apr 22, 2021, 6:41 PM • 1 Y
Y by Mango247
AB=AC=1
BC=4/sqrt(5)
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N Quick Reply
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