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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.

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[*]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
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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
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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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2000th Post!
PikaPika999   13
N 17 minutes ago by Dragon_Yang
1. How many ways can you arrange the letters in the word ALGEBRA such that no two identical letters are adjacent?

2. Find the smallest positive integer n such that $n^2+n+41$ is not a prime number.

3. You have 4 red tiles, 3 blue tiles, and 2 green tiles. How many ways can you arrange them in a row such that no two tiles of the same color are adjacent?

4. You flip a fair coin repeatedly until you either get 3 tails or 4 heads. What is the expected number of flips before stopping?

5. Let $A(2,3)$ and $B(8,7)$ be two points in the coordinate plane. A circle is drawn such that $\overline{AB}$ is a diameter.

(a). Find the equation of the circle in the form $(x+a)^2+(y+b)^2=r^2$

(b). A line passes through the point P(6,2) and is tangent to the circle. Find the equation of this tangent line.

hopefully these problems weren't too easy lol

also,
Please tell me if any of these problems have any flaws! (also please put your answers in hide tags or quote tags)
13 replies
PikaPika999
Today at 1:53 AM
Dragon_Yang
17 minutes ago
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Inequality on non-nagative numbers
TUAN2k8   0
20 minutes ago
Source: My book
Let $a,b,c$ be non-nagative real numbers such that $a+b+c=3$.
Prove that $ab+bc+ca-abc \leq \frac{9}{4}$.
0 replies
1 viewing
TUAN2k8
20 minutes ago
0 replies
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Incircle in an isoscoles triangle
Sadigly   2
N 35 minutes ago by Sadigly
Source: own
Let $ABC$ be an isosceles triangle with $AB=AC$, and let $I$ be its incenter. Incircle touches sides $BC,CA,AB$ at $D,E,F$, respectively. Foot of altitudes from $E,F$ to $BC$ are $X,Y$ , respectively. Rays $XI,YI$ intersect $(ABC)$ at $P,Q$, respectively. Prove that $(PQD)$ touches incircle at $D$.
2 replies
Sadigly
Friday at 9:21 PM
Sadigly
35 minutes ago
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Prove that the triangle is isosceles.
TUAN2k8   7
N an hour ago by TUAN2k8
Source: My book
Given acute triangle $ABC$ with two altitudes $CF$ and $BE$.Let $D$ be the point on the line $CF$ such that $DB \perp BC$.The lines $AD$ and $EF$ intersect at point $X$, and $Y$ is the point on segment $BX$ such that $CY \perp BY$.Suppose that $CF$ bisects $BE$.Prove that triangle $ACY$ is isosceles.
7 replies
1 viewing
TUAN2k8
May 16, 2025
TUAN2k8
an hour ago
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Locus of Mobile points on Circle and Square
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
Source: 2012 Hitotsubashi University entrance exam, problem 4
In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane.

(1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$,
find the equation of the locus of the point $R$.

(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.
1 reply
Kunihiko_Chikaya
Feb 28, 2012
Mathzeus1024
an hour ago
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Fun challange problem :)
TigerSenju   10
N 2 hours ago by Pengu14
Scenario:

Master Alchemist Aurelius is renowned for his mastery of elemental fusion. He works with seven fundamental, yet mysterious, elements: Ignis (Fire), Aqua (Water), Terra (Earth), Aer (Air), Lux (Light), Umbra (Shadow), and Aether (Spirit). Each element possesses a unique 'potency' value, a positive integer crucial for his most complex fusions

Aurelius has lost his master log of these potencies. All he has left are seven cryptic scrolls, each containing a precise relationship between the potencies of various elements. He needs these values to complete his Grand Device. Can you help him deduce the exact potency of each element?

The Elements and Their Potencies:

Let I represent the potency of Ignis (Fire).
Let A represent the potency of Aqua (Water).
Let T represent the potency of Terra (Earth).
Let R represent the potency of Aer (Air).
Let L represent the potency of Lux (Light).
Let U represent the potency of Umbra (Shadow).
Let E represent the potency of Aether (Spirit).
The Cryptic Scrolls (System of Equations):

Aurelius's scrolls reveal the following relationships:

The combined potency of Ignis, Aqua, and Terra is equal to the potency of Aer plus Lux, plus a constant of two.

If you sum the potencies of Aqua and Umbra, it precisely equals the sum of Lux and Aether, minus one.

The sum of Terra and Aer potencies is the same as the sum of Ignis, Aqua, and Aether potencies, minus one.

Three times the potency of Ignis, plus the potency of Aer, is equal to the sum of Aqua, Terra, and Aether potencies, plus five.

The difference between Lux and Ignis potencies is identical to the difference between Umbra and Aqua potencies.

The sum of Umbra and Aether potencies, when decreased by the potency of Terra, results in twice the potency of Aqua.

The potency of Ignis added to Lux, minus the potency of Aer, is equivalent to the potency of Aether minus Umbra, plus one.

The Grand Challenge:

Using only the information from the cryptic scrolls, set up and solve the system of seven linear equations to determine the unique positive integer potency value for each of the seven elements: I,A,T,R,L,U,E.

good luck, and whoever finds the potencies first, gets a title of The SYSTEMS OF EQUATIONS MASTER

p.s. Yes, I did just come up with a whole story of words to make a ridiculously long problem, but hey, you're reading this, so you probably have nothing better to be doing. ;)
10 replies
TigerSenju
Today at 1:04 AM
Pengu14
2 hours ago
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Circle is tangent to circumcircle and incircle
ABCDE   73
N 2 hours ago by AR17296174
Source: 2016 ELMO Problem 6
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.

(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.

(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$.

James Lin
73 replies
ABCDE
Jun 24, 2016
AR17296174
2 hours ago
J
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Mathematical Olympiad Finals 2013
parkjungmin   0
2 hours ago
Mathematical Olympiad Finals 2013
0 replies
parkjungmin
2 hours ago
0 replies
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n^k + mn^l + 1 divides n^(k+1) - 1
cjquines0   37
N 2 hours ago by alexanderhamilton124
Source: 2016 IMO Shortlist N4
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
37 replies
cjquines0
Jul 19, 2017
alexanderhamilton124
2 hours ago
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A very beautiful geo problem
TheMathBob   4
N 2 hours ago by ravengsd
Source: Polish MO Finals P2 2023
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$Prove that $YI$ is the angle bisector of $XYA$.
4 replies
TheMathBob
Mar 29, 2023
ravengsd
2 hours ago
J
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Difficult combinatorics problem
shactal   0
2 hours ago
Can someone help me with this problem? Let $n\in \mathbb N^*$. We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$. We then denote $A_i>A_j$. A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?
0 replies
shactal
2 hours ago
0 replies
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Cubic and Quadratic
mathisreal   3
N 2 hours ago by macves
Source: CIIM 2020 P2
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$
3 replies
mathisreal
Oct 26, 2020
macves
2 hours ago
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9 How many squares do you have memorized
LXC007   27
N Today at 4:53 AM by whwlqkd
How many squares have you memorized. I have 1-20
27 replies
LXC007
Yesterday at 3:44 PM
whwlqkd
Today at 4:53 AM
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Challenge: Make every number to 100 using 4 fours
CJB19   126
N Today at 4:15 AM by e_is_2.7182818
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:
$$(4-4)*4*4=0$$$$4-4+4/4=1$$$$4/4+4/4=2$$$$(4+4+4)/4=3$$$$4+(4-4)*4=4$$$$4+4^{4-4}=5$$$$4!/4+4-4=6$$$$4+4-4/4=7$$$$4+4+4-4=8$$
126 replies
CJB19
May 15, 2025
e_is_2.7182818
Today at 4:15 AM
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simplify
ngelyy   9
N Apr 25, 2025 by EthanNg6
$\frac{24x}{21}+\frac{35x}{49}-\frac{x}{2}$
9 replies
ngelyy
Apr 18, 2025
EthanNg6
Apr 25, 2025
J
simplify
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Reveal topic tags
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ngelyy
140 posts
ngelyy
#1 h  Apr 18, 2025, 2:59 AM
Y by
$\frac{24x}{21}+\frac{35x}{49}-\frac{x}{2}$
This post has been edited 1 time. Last edited by Eternica, Apr 27, 2025, 12:03 PM
Z K Y
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maxamc
581 posts
maxamc
#4 h  Apr 18, 2025, 3:04 AM
Y by
This question is impossible.
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ngelyy
140 posts
ngelyy
#6 h  Apr 18, 2025, 3:06 AM
Y by
maxamc wrote:
This question is impossible.

thank youu!!
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ngelyy
140 posts
ngelyy
#7 h  Apr 18, 2025, 3:13 AM
Y by
ive answered these 2 questions but i think i made a mistake

$3y+\frac{y-8}{2}+\frac{6y}{4}$

$\frac{20z-1}{3}-\frac{8z+4}{12}$

can anyone help me?
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UberPiggy
43 posts
UberPiggy
#8 h  Apr 18, 2025, 3:31 AM • 1 Y
Y by ngelyy
first one

second one
This post has been edited 1 time. Last edited by UberPiggy, Apr 18, 2025, 3:34 AM
Reason: added second problem
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maxamc
581 posts
maxamc
#9 h  Apr 18, 2025, 3:34 AM • 1 Y
Y by ngelyy
ngelyy wrote:
ive answered these 2 questions but i think i made a mistake

$3y+\frac{y-8}{2}+\frac{6y}{4}$

$\frac{20z-1}{3}-\frac{8z+4}{12}$

can anyone help me?

second one
Z K Y
The post below has been deleted. Click to close.
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ngelyy
140 posts
ngelyy
#10 h  Apr 18, 2025, 3:35 AM • 1 Y
Y by UberPiggy
oh i miscalculated the first one
thank you so much guys!! :thumbup:
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EthanNg6
35 posts
EthanNg6
#11 h  Apr 19, 2025, 2:25 AM
Y by
If you're solving for x, this question is impossible.
Z K Y
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K1mchi_
125 posts
K1mchi_
#13 h  Apr 19, 2025, 2:19 PM
Y by
EthanNg6 wrote:
If you're solving for x, this question is impossible.

“Simplify inequality”
Z K Y
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EthanNg6
35 posts
EthanNg6
#14 h  Apr 25, 2025, 11:41 PM
Y by
K1mchi_ wrote:
EthanNg6 wrote:
If you're solving for x, this question is impossible.

“Simplify inequality”

Click to reveal hidden text
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