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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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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 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 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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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   0
2 hours 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
2 hours ago
0 replies
J
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Diagonal of a pentagon that divides it into a triangle and a cyclic quadrilatera
EmersonSoriano   0
2 hours 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
2 hours ago
0 replies
J
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Eazy equation clap
giangtruong13   0
4 hours ago
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$$
0 replies
giangtruong13
4 hours ago
0 replies
J
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inequality
revol_ufiaw   3
N 5 hours ago by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
Today at 2:05 PM
MS_asdfgzxcvb
5 hours ago
J
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How to judge a number is prime or not?
mingzhehu   0
6 hours ago
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
0 replies
mingzhehu
6 hours ago
0 replies
J
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N 6 hours ago by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
6 hours ago
J
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J
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areas, regular hexagon and a square - 2021 COMATEQ p2 (Juniors)
parmenides51   4
N Jul 23, 2021 by Spakian
Suppose a regular hexagon and a square share a side as in the figure.
Let $A$ be the area of the hexagon and $B$ the area of the square. Calculate he quotient $A/B$.
IMAGE
4 replies
parmenides51
Jul 23, 2021
Spakian
Jul 23, 2021
J
areas, regular hexagon and a square - 2021 COMATEQ p2 (Juniors)
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Reveal topic tags
geometry
areas
square
hexagon
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parmenides51
30629 posts
parmenides51
#1 h  Jul 23, 2021, 3:39 PM • 1 Y
Y by HWenslawski
Suppose a regular hexagon and a square share a side as in the figure.
Let $A$ be the area of the hexagon and $B$ the area of the square. Calculate he quotient $A/B$.
https://cdn.artofproblemsolving.com/attachments/b/1/ec9853c0aa637ceada161f12c0f1745943fe6c.png
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Mathdreams
1448 posts
Mathdreams
#2 h  Jul 23, 2021, 3:44 PM • 1 Y
Y by parmenides51
Answer?
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Hellboy1234
134 posts
Hellboy1234
#3 h  Jul 23, 2021, 4:27 PM
Y by
note that if we join each vertex of a regular hexagon....we can get 6 equilateral triangles with side lengths = sides of that regular hexagon = $a$ (here)
so area of the ehxagon = $6. \frac{\sqrt{3}}{4}.a^2 = \frac{3.\sqrt{3}}{2}.a^2 =A$
area of the square = $a^2 =B$
so $$\frac{A}{B}= \frac{3.\sqrt{3}}{2}$$
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AOPqghj
546 posts
AOPqghj
#4 h  Jul 23, 2021, 4:34 PM
Y by
Hellboy1234 wrote:
note that if we join each vertex of a regular hexagon....we can get 6 equilateral triangles with side lengths = sides of that regular hexagon = $a$ (here)
so area of the ehxagon = $6\cdot\frac{\sqrt{3}}{4}\cdot a^2 = \frac{3\cdot \sqrt{3}}{2}\cdot a^2 =A$
area of the square = $a^2 =B$
so $$\frac{A}{B}= \frac{3\cdot \sqrt{3}}{2}$$

FTFY
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Spakian
304 posts
Spakian
#5 h  Jul 23, 2021, 6:17 PM
Y by
parmenides51 wrote:
Suppose a regular hexagon and a square share a side as in the figure.
Let $A$ be the area of the hexagon and $B$ the area of the square. Calculate he quotient $A/B$.
https://cdn.artofproblemsolving.com/attachments/b/1/ec9853c0aa637ceada161f12c0f1745943fe6c.png

Sol
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