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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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
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
Challenge: Make every number to 100 using 4 fours
CJB19   180
N 7 minutes ago by steve4916
I've seen this attempted a lot but I want to see if the AoPS community can actually do it. Using ONLY 4 fours and math operations, make as many numbers as you can. Try to go in order. I'll start:
$$(4-4)*4*4=0$$$$4-4+4/4=1$$$$4/4+4/4=2$$$$(4+4+4)/4=3$$$$4+(4-4)*4=4$$$$4+4^{4-4}=5$$$$4!/4+4-4=6$$$$4+4-4/4=7$$$$4+4+4-4=8$$
180 replies
CJB19
May 15, 2025
steve4916
7 minutes ago
9 How many squares do you have memorized
LXC007   44
N 15 minutes ago by Aaronjudgeisgoat
How many squares have you memorized. I have 1-20
44 replies
LXC007
Saturday at 3:44 PM
Aaronjudgeisgoat
15 minutes ago
Find digits
MihaiT   19
N 2 hours ago by Aaronjudgeisgoat
Find digits $a,b,c$ s.t. $\frac{a}{a+b+c}=\overline{0,abc }$
19 replies
MihaiT
Yesterday at 4:56 PM
Aaronjudgeisgoat
2 hours ago
Nice concurrency
navi_09220114   1
N 2 hours ago by bin_sherlo
Source: TASIMO 2025 Day 1 Problem 2
Four points $A$, $B$, $C$, $D$ lie on a semicircle $\omega$ in this order with diameter $AD$, and $AD$ is not parallel to $BC$. Points $X$ and $Y$ lie on segments $AC$ and $BD$ respectively such that $BX\parallel AD$ and $CY\perp AD$. A circle $\Gamma$ passes through $D$ and $Y$ is tangent to $AD$, and intersects $\omega$ again at $Z\neq D$. Prove that the lines $AZ$, $BC$ and $XY$ are concurrent.
1 reply
navi_09220114
2 hours ago
bin_sherlo
2 hours ago
Similar triangles and cyclic quadrilaterals
tapir1729   24
N 3 hours ago by Rayvhs
Source: TSTST 2024, problem 8
Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel.

Michael Ren
24 replies
tapir1729
Jun 24, 2024
Rayvhs
3 hours ago
Another triangle
Rushil   15
N 3 hours ago by lakshya2009
Source: Indian RMO 1991 Problem 1
Let $P$ be an interior point of a triangle $ABC$ and $AP,BP,CP$ meet the sides $BC,CA,AB$ in $D,E,F$ respectively. Show that \[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}.  \]
Remark
15 replies
Rushil
Oct 15, 2005
lakshya2009
3 hours ago
annoying exponent simplification. pls help
Miranda2829   5
N 3 hours ago by giratina3
(3xy-3x-6)² ( 2x²-3y²-3)²
--------------------------------------
(3xy-3x-6)⁵ (-x³-6xy+4x) ⁵

It is upper one divide lower part.. sorry, i can't get the right form
tips ..... Thanks
5 replies
Miranda2829
Yesterday at 9:56 PM
giratina3
3 hours ago
Point inside parallelogram
BigSams   21
N 3 hours ago by Want-to-study-in-NTU-MATH
Source: Canadian Mathematical Olympiad - 1997 - Problem 4.
The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.
21 replies
BigSams
May 7, 2011
Want-to-study-in-NTU-MATH
3 hours ago
Geometry
MathsII-enjoy   1
N 3 hours ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$ with $M$ being the midpoint of $BC$. The tangents at $B, C$ of $(O)$ intersect at $D$. Let $N$ be the projection of $O$ onto $AD$. On the perpendicular bisector of $BC$, take a point $K$ that is not on $(O)$ and different from M. Circle $(KBC)$ intersects $AK$ at $F$. Lines $NF$ and $AM$ intersect at $E$. Prove that $AEF$ is an isosceles triangle.
1 reply
MathsII-enjoy
May 15, 2025
MathsII-enjoy
3 hours ago
Probably a good lemma
Zavyk09   5
N 3 hours ago by Orzify
Source: found when solving exercises
Let $ABC$ be a triangle with circumcircle $\omega$. Arbitrary points $E, F$ on $AC, AB$ respectively. Circumcircle $\Omega$ of triangle $AEF$ intersects $\omega$ at $P \ne A$. $BE$ intersects $CF$ at $I$. $PI$ cuts $\Omega$ and $\omega$ at $K, L$ respectively. Construct parallelogram $KFRE$. Prove that $A, R, L$ are collinear.
5 replies
Zavyk09
Yesterday at 12:50 PM
Orzify
3 hours ago
Two perpendiculars
jayme   2
N 4 hours ago by jayme
Source: Own?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.

Prove : CV is perpendicular to BC.

Sincerely
Jean-Louis
2 replies
jayme
Today at 6:19 AM
jayme
4 hours ago
13rd ibmo - rep. dominicana 1998/q2.
carlosbr   6
N 5 hours ago by fearsum_fyz
Source: Spanish Communities
The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.
6 replies
carlosbr
Apr 16, 2006
fearsum_fyz
5 hours ago
RMM 2013 Problem 6
dr_Civot   15
N 6 hours ago by N3bula
A token is placed at each vertex of a regular $2n$-gon. A move consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
15 replies
dr_Civot
Mar 3, 2013
N3bula
6 hours ago
Distance between any two points is irrational
orl   21
N Today at 6:35 AM by cursed_tangent1434
Source: IMO 1987, Day 2, Problem 5
Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
21 replies
orl
Nov 11, 2005
cursed_tangent1434
Today at 6:35 AM
Counting Problems
mithu542   5
N May 7, 2025 by BS2012
Hello!

Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator) hint


1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.

2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.

3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.

4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.

5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.

6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.

7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.

8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.

9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.

10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
5 replies
mithu542
Apr 28, 2025
BS2012
May 7, 2025
Counting Problems
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mithu542
1584 posts
#1 • 2 Y
Y by PikaPika999, Exponent11
Hello!

Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator) hint


1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.

2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.

3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.

4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.

5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.

6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.

7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.

8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.

9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.

10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
This post has been edited 3 times. Last edited by mithu542, Apr 29, 2025, 9:30 PM
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Bummer12345
150 posts
#2 • 2 Y
Y by PikaPika999, Exponent11
number 1 has to be inspired by that one target p8 question with 6 dice and sum to 7
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Inaaya
393 posts
#3 • 2 Y
Y by PikaPika999, Exponent11
ill solve some of these when i get some of my math and ai4girls done trust
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Math-lover1
301 posts
#4 • 2 Y
Y by PikaPika999, Exponent11
Problem 10 doesn't make sense since each face has 4 edges adjacent to it, not 3.
However, each vertex has 3 edges adjacent to it. If we're considering vertices...

solution to P10 if considering vertices

I might be a bit too late for this one :P
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Math-lover1
301 posts
#5 • 2 Y
Y by PikaPika999, Exponent11
S9
This post has been edited 1 time. Last edited by Math-lover1, May 6, 2025, 6:39 PM
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BS2012
1047 posts
#6 • 3 Y
Y by PikaPika999, Exponent11, MathPerson12321
Math-lover1 wrote:
Problem 10 doesn't make sense since each face has 4 edges adjacent to it, not 3.
However, each vertex has 3 edges adjacent to it. If we're considering vertices...

solution to P10 if considering vertices

I might be a bit too late for this one :P

This is incorrect. In total, there are $3^{12}$ ways to color the edges, so the denominator of the answer, in lowest terms, should divide $3^{12}$ because the probability is the number of successful outcomes over the number of possible outcomes. We have that $9^8=3^{16}$ does not divide $3^{12}.$

In general, linearity of expectation only works for adding expectations, not multiplying them. For example, it is not generally true that $E(XY)=E(X)E(Y)$ for variables $X$ and $Y$ that are not independent.

I think this problem can be done by casework on the colors of the edges on the sides but that seems kinda messy
This post has been edited 5 times. Last edited by BS2012, May 7, 2025, 12:47 AM
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