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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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jlacosta
Apr 2, 2025
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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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Geometry Proof
Jackson0423   0
18 minutes ago
In triangle \( \triangle ABC \), point \( P \) on \( AB \) satisfies \( DB = BC \) and \( \angle DCA = 30^\circ \).
Let \( X \) be the point where the perpendicular from \( B \) to line \( DC \) meets the angle bisector of \( \angle BCA \).
Then, the relation \( AD \cdot DC = BD \cdot AX \) holds.

Prove that \( \triangle ABC \) is an isosceles triangle.
0 replies
Jackson0423
18 minutes ago
0 replies
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a nice prob for number theory
Jackson0423   0
20 minutes ago
Source: number theory
Let \( n \) be a positive integer, and let its positive divisors be
\[ d_1 < d_2 < \cdots < d_k. \]Define \( f(n) \) to be the number of ordered pairs \( (i, j) \) with \( 1 \le i, j \le k \) such that \( \gcd(d_i, d_j) = 1 \).

Find \( f(3431 \times 2999) \).

Also, find a general formula for \( f(n) \) when
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}, \]where the \( p_i \) are distinct primes and the \( e_i \) are positive integers.
0 replies
Jackson0423
20 minutes ago
0 replies
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Very easy NT
GreekIdiot   6
N 21 minutes ago by ektorasmiliotis
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
6 replies
GreekIdiot
2 hours ago
ektorasmiliotis
21 minutes ago
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Functionnal equation
Rayanelba   0
25 minutes ago
Source: Own
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(f(x)))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
0 replies
Rayanelba
25 minutes ago
0 replies
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Great sequence problem
Assassino9931   1
N 30 minutes ago by internationalnick123456
Source: Balkan MO Shortlist 2024 N4
Let $k$ be a positive integer. Determine all sequences $(a_n)_{n\geq 1}$ of positive integers such that
$$ a_{n+2}(a_{n+1} - k) = a_n(a_{n+1} + k) $$for all positive integers $n$.
1 reply
Assassino9931
Apr 27, 2025
internationalnick123456
30 minutes ago
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INMO 2018 -- Problem #3
integrated_JRC   44
N 34 minutes ago by bjump
Source: INMO 2018
Let $\Gamma_1$ and $\Gamma_2$ be two circles with respective centres $O_1$ and $O_2$ intersecting in two distinct points $A$ and $B$ such that $\angle{O_1AO_2}$ is an obtuse angle. Let the circumcircle of $\Delta{O_1AO_2}$ intersect $\Gamma_1$ and $\Gamma_2$ respectively in points $C (\neq A)$ and $D (\neq A)$. Let the line $CB$ intersect $\Gamma_2$ in $E$ ; let the line $DB$ intersect $\Gamma_1$ in $F$. Prove that, the points $C, D, E, F$ are concyclic.
44 replies
integrated_JRC
Jan 21, 2018
bjump
34 minutes ago
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Queue geo
vincentwant   0
41 minutes ago
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
0 replies
vincentwant
41 minutes ago
0 replies
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Linear colorings mod 2^n
vincentwant   0
42 minutes ago
Let $n$ be a positive integer. The ordered pairs $(x,y)$ where $x,y$ are integers in $[0,2^n)$ are each labeled with a positive integer less than or equal to $2^n$ such that every label is used exactly $2^n$ times and there exist integers $a_1,a_2,\dots,a_{2^n}$ and $b_1,b_2,\dots,b_{2^n}$ such that the following property holds: For any two lattice points $(x_1,y_1)$ and $(x_2,y_2)$ that are both labeled $t$, there exists an integer $k$ such that $x_2-x_1-ka_t$ and $y_2-y_1-kb_t$ are both divisible by $2^n$. How many such labelings exist?
0 replies
vincentwant
42 minutes ago
0 replies
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sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   0
44 minutes ago
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases} \sqrt{n}&\text{ if n is a perfect square} \\ n+p&\text{ otherwise} \end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
0 replies
vincentwant
44 minutes ago
0 replies
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Reducibility of 2x^2 cyclotomic
vincentwant   0
an hour ago
Let $S$ denote the set of all positive integers less than $1020$ that are relatively prime to $1020$. Let $\omega=\cos\frac{\pi}{510}+i\sin\frac{\pi}{510}$. Is the polynomial $$\prod_{n\in S}(2x^2-\omega^n)$$reducible over the rational numbers, given that it has integer coefficients?
0 replies
vincentwant
an hour ago
0 replies
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Basic geometry
AlexCenteno2007   6
N 2 hours ago by vanstraelen
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
6 replies
AlexCenteno2007
Feb 9, 2025
vanstraelen
2 hours ago
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Algebraic Manipulation
Darealzolt   1
N 2 hours ago by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
3 hours ago
Soupboy0
2 hours ago
J
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BrUMO 2025 Team Round Problem 13
lpieleanu   1
N 2 hours ago by vanstraelen
Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
2 hours ago
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Inequlities
sqing   33
N 3 hours ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
3 hours ago
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trapezoid area given 4 sidelengths (1995 Denmark MO - Mohr Contest p1)
parmenides51   14
N Aug 18, 2020 by SilverDragon246
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.IMAGE
14 replies
parmenides51
Aug 18, 2020
SilverDragon246
Aug 18, 2020
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trapezoid area given 4 sidelengths (1995 Denmark MO - Mohr Contest p1)
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Reveal topic tags
geometry
trapezoid
area of a triangle
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parmenides51
30650 posts
parmenides51
#1 h  Aug 18, 2020, 7:38 PM • 2 Y
Y by Mango247, Mango247
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.//cdn.artofproblemsolving.com/images/2/2/6/22682ac80ee2fa6ef0835aa0c6538f37c4030e71.png
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natmath
8219 posts
natmath
#2 h  Aug 18, 2020, 7:56 PM
Y by
If x and y are the "leftover" lengths on the 36 base, we have
x+y=25
625-x^2=900-y^2

(y-x)(y+x)=275
y-x=11
y=18
x=7
So $h=24$ and the area is $47*12$
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franzliszt
23531 posts
franzliszt
#3 h  Aug 18, 2020, 7:57 PM
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Darn how are you so fast??

Solution
This post has been edited 2 times. Last edited by franzliszt, Aug 18, 2020, 8:00 PM
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OlympusHero
17020 posts
OlympusHero
#4 h  Aug 18, 2020, 8:03 PM • 3 Y
Y by Mango247, Mango247, Mango247
parmenides51 wrote:
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png

We would like to use Pythagorean Triples. A height of $24$ seems to work with 7-24-25 and 18-24-30. This checks out so $\frac{(11+36) \cdot 24}{2}=\boxed{564}$.
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parmenides51
30650 posts
parmenides51
#5 h  Aug 18, 2020, 8:18 PM
Y by
exists another solution without pythagorean :)
This post has been edited 2 times. Last edited by parmenides51, Aug 18, 2020, 8:24 PM
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IAmTheHazard
5001 posts
IAmTheHazard
#6 h  Aug 18, 2020, 8:21 PM
Y by
Well the formula for a trapezoid with sides $a$ and $b$ parallel is $\frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)}$, so you could use that.
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OlympusHero
17020 posts
OlympusHero
#7 h  Aug 18, 2020, 8:26 PM
Y by
There's also a general formula for the area of a quadrilateral, right? Maybe I misremember, but in the challenge problems from precalculus chapter 4 I think there is a general formula for the area of a quadrilateral.
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franzliszt
23531 posts
franzliszt
#8 h  Aug 18, 2020, 8:26 PM
Y by
OlympusHero wrote:
There's also a general formula for the area of a quadrilateral, right? Maybe I misremember, but in the challenge problems from precalculus chapter 4 I think there is a general formula for the area of a quadrilateral.

I thought it had to be cyclic?
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IAmTheHazard
5001 posts
IAmTheHazard
#9 h  Aug 18, 2020, 8:31 PM • 1 Y
Y by Mango247
There's a general formula but you need some angles. It's called Bretschnider's or something.
Cyclic is brahmagupta.
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parmenides51
30650 posts
parmenides51
#10 h  Aug 18, 2020, 8:57 PM
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from https://en.wikipedia.org/wiki/QuadrilateralClick to reveal hidden text
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franzliszt
23531 posts
franzliszt
#11 h  Aug 18, 2020, 8:58 PM
Y by
I honestly would rather use Pythag bash than Bretschneider's :)
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IAmTheHazard
5001 posts
IAmTheHazard
#12 h  Aug 18, 2020, 9:03 PM
Y by
I don't think bretschneider's alone works here since w/o pythagoras you don't know the angles, and if you do work to find the angles then you've already found the height.
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parmenides51
30650 posts
parmenides51
#13 h  Aug 18, 2020, 9:03 PM • 1 Y
Y by Mango247
parmenides51 wrote:
exists another solution without pythagorean :)

my solution
1st hint, why?, and because
2nd hint ,the answer is
This post has been edited 2 times. Last edited by parmenides51, Aug 18, 2020, 9:05 PM
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PM.MATHLEAGUE.VA.CHAMP
738 posts
PM.MATHLEAGUE.VA.CHAMP
#14 h  Aug 18, 2020, 9:48 PM
Y by
I thought the general area formula for a quadrilateral was $\sqrt{(s-a)(s-b)(s-c)(s-d)}$
This post has been edited 1 time. Last edited by PM.MATHLEAGUE.VA.CHAMP, Aug 18, 2020, 9:49 PM
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SilverDragon246
678 posts
SilverDragon246
#15 h  Aug 18, 2020, 10:00 PM
Y by
@Above that is only if the quadrilateral is cyclic. As post #10 said, the general area formula of a quadrilateral is given as $$ {\displaystyle {\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;[1+\cos(A+C)]}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\left[\cos ^{2}\left({\tfrac {A+C}{2}}\right)\right]}}\end{aligned}}}{\begin{aligned}\end{aligned}}$$
but in cyclic quadrilaterals, we know that opposite angles add up to 180, and since $\cos(180^{\circ}) = -1$, the formula then simplifies down to just $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ (the last term cancels out). But this only works for cyclic quadrilaterals.
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