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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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primes,exponentials,factorials
skellyrah   3
N an hour ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
3 replies
1 viewing
skellyrah
4 hours ago
skellyrah
an hour ago
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Transformation of a cross product when multiplied by matrix A
Math-lover1   2
N an hour ago by Math-lover1
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$, but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?
2 replies
Math-lover1
Yesterday at 10:29 PM
Math-lover1
an hour ago
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af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
shoki   26
N an hour ago by MathLuis
Source: Iran TST 2011 - Day 4 - Problem 3
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
26 replies
shoki
May 14, 2011
MathLuis
an hour ago
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Very easy NT
GreekIdiot   8
N an hour ago by vsamc
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
8 replies
GreekIdiot
Today at 2:49 PM
vsamc
an hour ago
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Another quadrilateral in a circle
v_Enhance   110
N an hour ago by Marco22
Source: APMO 2013, Problem 5
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
110 replies
v_Enhance
May 3, 2013
Marco22
an hour ago
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Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   64
N an hour ago by lpieleanu
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
64 replies
v_Enhance
Jul 18, 2014
lpieleanu
an hour ago
J
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unfair coin, points winning 2024 TMC AIME Mock #9
parmenides51   5
N an hour ago by Math-lover1
Krithik has an unfair coin with a $\frac13$ chance of landing heads when flipped. Krithik is playing a game where he starts with $1$ point. Every turn, he flips the coin, and if it lands heads, he gains $1$ point, and if it lands tails, he loses $1$ point. However, after the turn, if he has a negative number of points, his point counter resets to $1$. Krithik wins when he earns $8$ points. Find the expected number of turns until Krithik wins.
5 replies
+1 w
parmenides51
Apr 26, 2025
Math-lover1
an hour ago
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Queue geo
vincentwant   2
N 2 hours ago by MathLuis
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 $(KPQ)$.
2 replies
vincentwant
Today at 3:54 PM
MathLuis
2 hours ago
J
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Functional Geometry
GreekIdiot   2
N 2 hours ago by Double07
Source: BMO 2024 SL G7
Let $f: \pi \to \mathbb R$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $\Delta ABC$ with circumcenter $O$, centroid $G$ and orthocenter $H$. Prove that $f$ is constant.
2 replies
GreekIdiot
Apr 27, 2025
Double07
2 hours ago
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Right-angled triangle if circumcentre is on circle
liberator   78
N 2 hours ago by bin_sherlo
Source: IMO 2013 Problem 3
Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

Proposed by Alexander A. Polyansky, Russia
78 replies
liberator
Jan 4, 2016
bin_sherlo
2 hours ago
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Can you construct the incenter of a triangle ABC?
PennyLane_31   3
N 2 hours ago by cj13609517288
Source: 2023 Girls in Mathematics Tournament- Level B, Problem 4
Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.
3 replies
PennyLane_31
Oct 29, 2023
cj13609517288
2 hours ago
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BrUMO 2025 Team Round Problem 15
lpieleanu   1
N 2 hours ago by vanstraelen
Let $\triangle{ABC}$ be an isosceles triangle with $AB=AC.$ Let $D$ be a point on the circumcircle of $\triangle{ABC}$ on minor arc $AB.$ Let $\overline{AD}$ intersect the extension of $\overline{BC}$ at $E.$ Let $F$ be the midpoint of segment $AC,$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AB}.$ Let the extension of $\overline{DG}$ intersect $\overline{AC}$ and the circumcircle of $\triangle{ABC}$ at $H$ and $I,$ respectively. Given that $DG=3, GH=5,$ and $HI=1,$ compute the length of $\overline{AE}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
2 hours ago
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Do not try to bash on beautiful geometry
ItzsleepyXD   3
N 2 hours ago by FarrukhBurzu
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
3 replies
ItzsleepyXD
Today at 9:30 AM
FarrukhBurzu
2 hours ago
J
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trigonometric functions
VivaanKam   9
N 3 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Yesterday at 8:29 PM
aok
3 hours ago
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J
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(PAB) + (PCD) > (PBC) + (PDA) trapezoid 2010 Estonia Open Senior 2.1
parmenides51   1
N Oct 4, 2020 by natmath
The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.
1 reply
parmenides51
Oct 4, 2020
natmath
Oct 4, 2020
J
(PAB) + (PCD) > (PBC) + (PDA) trapezoid 2010 Estonia Open Senior 2.1
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Reveal topic tags
geometry
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parmenides51
30650 posts
parmenides51
#1 h  Oct 4, 2020, 6:46 PM
Y by
The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.
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natmath
8219 posts
natmath
#2 h  Oct 4, 2020, 9:07 PM
Y by
Redefine the variables, so that we are trying to prove the inequality $ab>cd$

It's a well known fact $ab=cd=K$ and that $c=d=\sqrt{K}$. So $b=\frac{K}{a}$

Use $AM-GM$ on $a+\frac{K}{a}$ to get the desired inequality of $a+\frac{K}{a}\geq2\sqrt{K}$. Equality is attained when $a=b$ which occurs when $ABCD$ is a parallelogram. Since a parallelogram is not a trapezoid, the inequality never attains an equality condition.
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