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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.

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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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Interesting inequality
sqing   4
N 25 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$ a^4+ b^4+c^4+6abc\leq9$$$$ a^3+ b^3+ c^3+3( \sqrt{3}-1)abc\leq 3\sqrt 3$$
4 replies
sqing
Yesterday at 2:54 AM
sqing
25 minutes ago
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Parameter and 4 variables
mihaig   0
25 minutes ago
Source: Own
Find the positive real constants $K$ such that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^K\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
0 replies
+1 w
mihaig
25 minutes ago
0 replies
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circumcenter of ARS lies on AD
Melid   0
28 minutes ago
Source: own
In triangle $ABC$, let $D$ be a point on arc $BC$ of circle $ABC$ which doesn't contain $A$. $AD$ and $BC$ intersect at $E$. Let $P$ and $Q$ be the reflection of $E$ about to $AB$ and $AC$, respectively. $PD$ intersects $AB$ at $R$, and $QD$ intersects $AC$ at $S$. Prove that circumcenter of triangle $ARS$ lies on $AD$.
0 replies
Melid
28 minutes ago
0 replies
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A sharp one with 3 var (3)
mihaig   6
N 37 minutes ago by mihaig
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
6 replies
mihaig
May 27, 2025
mihaig
37 minutes ago
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geometry problem with many circumcircles
Melid   1
N an hour ago by Beelzebub
Source: own
In scalene triangle $ABC$, which doesn't have right angle, let $O$ be its circumcenter. Circle $BOC$ intersects $AB$ and $AC$ at $A_{1}$ and $A_{2}$ for the second time, respectively. Similarly, circle $COA$ intersects $BC$ and $BA$ at $B_{1}$ and $B_{2}$, and circle $AOB$ intersects $CA$ and $CB$ at $C_{1}$ and $C_{2}$ for the second time, respectively. Let $O_{1}$ and $O_{2}$ be circumcenters of triangle $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$, respectively. Prove that $O, O_{1}, O_{2}$ are collinear.
1 reply
Melid
4 hours ago
Beelzebub
an hour ago
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weird conditions in geo
Davdav1232   3
N an hour ago by lakshya2009
Source: Israel TST 7 2025 p1
Let \( \triangle ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) be a point on \( AC \). Let \( L \) be a point inside the triangle such that \( \angle CLD = 90^\circ \) and
\[ CL \cdot BD = BL \cdot CD. \]Prove that the circumcenter of triangle \( \triangle BDL \) lies on line \( AB \).
3 replies
Davdav1232
May 8, 2025
lakshya2009
an hour ago
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Problem 4
codyj   88
N 2 hours ago by ND_
Source: IMO 2015 #4
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Proposed by Greece
88 replies
codyj
Jul 11, 2015
ND_
2 hours ago
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pairs (m, n) such that a fractional expression is an integer
cielblue   3
N 2 hours ago by Pal702004
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
3 replies
cielblue
May 24, 2025
Pal702004
2 hours ago
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interesting geometry config (3/3)
Royal_mhyasd   0
3 hours ago
Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$. If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$, prove that the intersection of lines $CT$ and $BS$ lies on $HE$.

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
0 replies
Royal_mhyasd
3 hours ago
0 replies
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Worst math problems
LXC007   50
N 3 hours ago by Yiyj
What is the most egregiously bad problem or solution you have encountered in school?
50 replies
LXC007
May 21, 2025
Yiyj
3 hours ago
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A perverse one
darij grinberg   7
N 3 hours ago by ezpotd
Source: German TST 2004, IMO ShortList 2003, number problem 2
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:

(i) move the last digit of $a$ to the first position to obtain the numb er $b$;
(ii) square $b$ to obtain the number $c$;
(iii) move the first digit of $c$ to the end to obtain the number $d$.

(All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)

Find all numbers $a$ for which $d\left( a\right) =a^2$.

Proposed by Zoran Sunic, USA
7 replies
darij grinberg
May 18, 2004
ezpotd
3 hours ago
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Overly wordy problems
ZMB038   16
N 3 hours ago by Yiyj
Hey everyone, here we can post questions with way to many extraneous words, that are actually easy.
Try to solve the one above yours.
I'll start:
Click to reveal hidden text
16 replies
ZMB038
May 28, 2025
Yiyj
3 hours ago
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Easy Probability Question (Optional Hardcore Question)
PikaVee   7
N 4 hours ago by PikaVee
(Thanks Random Stranger for the idea and I will be making it so it is extremely specific to your solution.)
We are playing Pokemon Scarlet and Violet and you are fighting a friend. You and your friend don't have any items at all and the pokemon does not have any held items. Your friend challenges you to a battle because he just said nah I'd win.

You two start the battle using only one Pokemon each which neither of you knows the type of the other. Luckily he had used a level 28 Squirtle and you had used a level 25 Pikachu. Surprisingly both of the Pokemon each have one HP. Your Pikachu has a move set of one single move of Thunder with 10/10 PP and has 1 HP because you forgot to go to the Pokemon center. Your Pikachu also has a bad IV stat in speed with 1/15 and the 252 EV speed stat of the Squirtle combined with a perfect IV stat in speed makes it so it guarantees to always out speed your move. To account for that he made his Squirtle have 1 HP on purpose for absolutely no reason.

After he saw what kind of moves you have and since that person was so cocky and confident that they decided to gamble all their moves with each having an equal chance of being used. Their Squirtle has a move set of Protect 10 PP which has 100% chance of being used and has has the success probability multiplied by 1/3 every time it is being used (Meaning the second time it is being used has a 33% chance of succeeding and a third time it will be 11%. This also ignores the rules of how the move is regularly used by making the 4th move 1/27 instead of it being a guaranteed fail and so on.), Tackle which has a 100% chance to hit having 10/35 PP , Water Gun which has a 100% chance of hitting with 10/25 and Rain Dance with 5/5 PP and 100% chance of being used. If the amount of PP reaches 0 it will be unavailable for the rest of the fight meaning that the probability for each other move to be used goes from 25% all the way to 33%.

For everyone who wants to solve the easy part. If the probability that Squirtle will survive turn 1 when simplified is a/b then what is a+b?

Alright so for Squirtle to survive turn one then we try to find out how Squirte will faint at turn one. First of all Pikachu needs to hit Thunder bolt at a 70% chance then get a 25% chance that the Squirtle will use Rain Dance so that the Squirtle will not faint the Pikachu because it didn't attack. Another possible option is for it to choose Water Gun and miss so it would be a 70% * 25% * 5% chance for Pikachu to faint Squirtle. 70% = \frac{7}{10}, 25% = \frac{1}{4}, and 5% = \frac{1}{20} So the complement of what we are trying to find is $\frac{7}{10}*\frac{1}{4}+\frac{7}{10}*\frac{1}{4}*\frac{1}{20}=\frac{7}{10*4}+\frac{7}{10*4*20}=\frac{7}{40}+\frac{7}{800}=\frac{140}{800}+\frac{7}{800}=\frac{147}{800}$. The complement of this would be $1-\frac{147}{800}$ or $\frac{653}{800}$. The final thing we can do is to make sure it is simplified and add the numerator and the denominator which is $653$ and $800$ so $653+800=1453$ This should be final answer. (Mathdash rating 800)

For the hard working fella that try-harded, What is the probability that the Squirtle will win this fight? (This is going to be a very long arithmetic series with a lot of cases. The max amount of turns this fight can have is 11 turns.)
7 replies
PikaVee
May 28, 2025
PikaVee
4 hours ago
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prime numbers
wpdnjs   128
N Today at 3:44 AM by Yihangzh
does anyone know how to quickly identify prime numbers?

thanks.
128 replies
wpdnjs
Oct 2, 2024
Yihangzh
Today at 3:44 AM
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J
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A geometry problem
Deomad123   2
N Apr 9, 2025 by Apple_maths60
Let $ABCD$ be an cyclic quadrilateral with $AB=8cm$,$BC=7cm$,$CD=6cm$ and $DA=5cm$ Find:$\frac{AC}{BD}$
2 replies
Deomad123
Apr 8, 2025
Apple_maths60
Apr 9, 2025
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A geometry problem
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Reveal topic tags
geometry
cyclic quadrilateral
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Deomad123
6 posts
Deomad123
#1 h  Apr 8, 2025, 7:20 PM
Y by
Let $ABCD$ be an cyclic quadrilateral with $AB=8cm$,$BC=7cm$,$CD=6cm$ and $DA=5cm$ Find:$\frac{AC}{BD}$
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exp-ipi-1
1074 posts
exp-ipi-1
#2 h  Apr 9, 2025, 1:33 AM
Y by
Let the intersection of $BD$ and $AC$ be $X$. Let the length of $DX=x$. Since triangles $ABX$ and $DCX$ are similar,$$\frac{DC}{AB}=\frac{DX}{AX}=\frac{6}{8},$$and it follows that $AX=\frac{4x}{3}$. Using similar triangles again, we get $BX=\frac{7AX}{5}=\frac{28x}{15}$ and $CX=\frac{3BX}{4}=\frac{21x}{15}$. $$\frac{AC}{BD}=\frac{\frac{4}{3}+\frac{21}{15}}{1+\frac{28}{15}}=\boxed{\frac{41}{43}}$$
This post has been edited 2 times. Last edited by exp-ipi-1, Apr 9, 2025, 1:33 AM
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Apple_maths60
27 posts
Apple_maths60
#3 h  Apr 9, 2025, 3:11 AM • 1 Y
Y by EthanNg6
AC/BD =((AB×AD)+(BC×CD))/((AB×BC)+(CD×AD))
= 8×5+7×6 / 8×7+5×6 =41/43 (ANSWER)
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