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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
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
Inequalities
sqing   1
N an hour ago by DAVROS
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
1 reply
+1 w
sqing
Yesterday at 2:23 PM
DAVROS
an hour ago
Maximum number of empty squares
Ecrin_eren   1
N an hour ago by Ecrin_eren


There are 16 kangaroos on a giant 4×4 chessboard, with exactly one kangaroo on each square. In each round, every kangaroo jumps to a neighboring square (up, down, left, or right — but not diagonally). All kangaroos stay on the board. More than one kangaroo can occupy the same square. What is the maximum number of empty squares that can exist after 100 rounds?



1 reply
Ecrin_eren
Yesterday at 6:35 PM
Ecrin_eren
an hour ago
Max and min of ab+bc+ca-abc
Tiira   6
N 3 hours ago by MathsII-enjoy
a, b and c are three non-negative reel numbers such that a+b+c=1.
What are the extremums of
ab+bc+ca-abc
?
6 replies
Tiira
Jan 29, 2021
MathsII-enjoy
3 hours ago
Polynomial Minimization
ReticulatedPython   4
N Today at 1:27 AM by jasperE3
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
4 replies
ReticulatedPython
May 6, 2025
jasperE3
Today at 1:27 AM
prove triangles are similar
N.T.TUAN   58
N Monday at 2:55 PM by mathwiz_1207
Source: USA Team Selection Test 2007, Problem 5
Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T = BT = C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.
58 replies
N.T.TUAN
Dec 8, 2007
mathwiz_1207
Monday at 2:55 PM
RMM 2013 Problem 6
dr_Civot   15
N Monday at 8:09 AM 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
Monday at 8:09 AM
incircle excenter midpoints
danepale   9
N May 18, 2025 by Want-to-study-in-NTU-MATH
Source: Middle European Mathematical Olympiad T-6
Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.
9 replies
danepale
Sep 21, 2014
Want-to-study-in-NTU-MATH
May 18, 2025
Three mutually tangent circles
math154   8
N May 17, 2025 by lakshya2009
Source: ELMO Shortlist 2011, G2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.

David Yang.
8 replies
math154
Jul 3, 2012
lakshya2009
May 17, 2025
Line AT passes through either S_1 or S_2
v_Enhance   89
N May 17, 2025 by zuat.e
Source: USA December TST for 57th IMO 2016, Problem 2
Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$.

Proposed by Evan Chen
89 replies
v_Enhance
Dec 21, 2015
zuat.e
May 17, 2025
Eight-point cicle
sandu2508   15
N May 16, 2025 by Mamadi
Source: Balkan MO 2010, Problem 2
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
15 replies
sandu2508
May 4, 2010
Mamadi
May 16, 2025
RMM 2013 Problem 3
dr_Civot   79
N May 16, 2025 by Ilikeminecraft
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
79 replies
dr_Civot
Mar 2, 2013
Ilikeminecraft
May 16, 2025
Hard Inequality
Asilbek777   2
N May 14, 2025 by Ritwin
Waits for Solution
2 replies
Asilbek777
May 14, 2025
Ritwin
May 14, 2025
ISI UGB 2025 P7
SomeonecoolLovesMaths   12
N May 14, 2025 by ohiorizzler1434
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
12 replies
SomeonecoolLovesMaths
May 11, 2025
ohiorizzler1434
May 14, 2025
angles in triangle
AndrewTom   34
N May 13, 2025 by happypi31415
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
34 replies
AndrewTom
Feb 1, 2013
happypi31415
May 13, 2025
trigonometric functions
VivaanKam   16
N May 16, 2025 by Shan3t
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
16 replies
VivaanKam
Apr 29, 2025
Shan3t
May 16, 2025
trigonometric functions
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G H BBookmark kLocked kLocked NReply
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VivaanKam
167 posts
#1 • 2 Y
Y by PikaPika999, linjiah
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
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Lijin
225 posts
#2 • 2 Y
Y by PikaPika999, linjiah
Are you talking about graphing them or just the basic ratios?
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Yiyj1
1266 posts
#4 • 2 Y
Y by PikaPika999, linjiah
Basic ratios: draw a right triangle. I'm terrible at asy so i can't draw one D:. Anyways, label one of the angles $\theta$. Then, label the hypotenuse with $H$, the leg adjacent to $\theta$ as $A$ (for adjacent) and the other leg $O$ (for opposite). Then, just remember this: SOH CAH TOA:\[\sin\theta=\dfrac{O}{H}, \cos\theta=\dfrac{A}{H}, \tan\theta=\dfrac{O}{A}.\]Then there are like $\sec, \csc, \cot$, which are the reciprocals of $\cos, \sin, \tan$. IMPORTANT: $\sec$ is the reciprocal of $\cos$ and $\csc$ is the reciprocal of $\sin$, not the other way around.
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aok
352 posts
#5 • 1 Y
Y by linjiah
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$
This post has been edited 4 times. Last edited by aok, Apr 29, 2025, 10:48 PM
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VivaanKam
167 posts
#6 • 1 Y
Y by linjiah
Yiyj1 wrote:
Basic ratios: draw a right triangle. I'm terrible at asy so i can't draw one D:. Anyways, label one of the angles $\theta$. Then, label the hypotenuse with $H$, the leg adjacent to $\theta$ as $A$ (for adjacent) and the other leg $O$ (for opposite). Then, just remember this: SOH CAH TOA:\[\sin\theta=\dfrac{O}{H}, \cos\theta=\dfrac{A}{H}, \tan\theta=\dfrac{O}{A}.\]Then there are like $\sec, \csc, \cot$, which are the reciprocals of $\cos, \sin, \tan$. IMPORTANT: $\sec$ is the reciprocal of $\cos$ and $\csc$ is the reciprocal of $\sin$, not the other way around.

So like this?

[asy]

draw((0,0)--(3,0)--(0,2)--cycle);
label("$\theta$", (2.7,0.1),W);
label("$A$", (1.5,0), S);
label("$O$", (0,1.205), W);
label("$H$", (1.2,1.1), NE);
[/asy]
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VivaanKam
167 posts
#7 • 1 Y
Y by linjiah
That’s cool! So if you have the lengths of a triangle you can find its angles?
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VivaanKam
167 posts
#8 • 1 Y
Y by linjiah
aok wrote:
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$

are they like polar quardinits ?
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VivaanKam
167 posts
#9 • 1 Y
Y by linjiah
but the wouldn't $\cos x$ have 2 values because on a circle there are two quordinates with the same $x$ position?
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lpieleanu
3001 posts
#10 • 1 Y
Y by linjiah
Yes, you can find the side lengths of a triangle given its angles. (If it is right, you can just use the standard ratio definitions of $\sin, \cos, \tan$ and use inverse trigonometric functions, and if it is not right, then you can use the Law of Cosines to find each angle.)

The point in rectangular coordinates $(\cos(\theta), \sin(\theta))$ corresponds to the point in polar coordinates $(1, \theta),$ i.e. $(\cos(\theta), \sin(\theta))$ is the point on the unit circle at an angle of $\theta$ radians counterclockwise of the positive $x$-axis.

Yes, the equation $\cos(x)=a$ has two solutions in $[0, 2\pi)$ for all $-1<a<1.$

Also, reminder that you can combine all of your questions into the same post. :)
This post has been edited 1 time. Last edited by lpieleanu, Apr 30, 2025, 6:39 PM
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aok
352 posts
#11 • 1 Y
Y by linjiah
that is correct, cos x = a has 2 solutions (generally)
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aok
352 posts
#12 • 1 Y
Y by linjiah
for x btw
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aok
352 posts
#13 • 1 Y
Y by linjiah
VivaanKam wrote:
That’s cool! So if you have the lengths of a triangle you can find its angles?

Correct, use the opposite of those functions.
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aok
352 posts
#14 • 1 Y
Y by linjiah
*use the cos theorem to find cos(x) then use the cos^-1
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BlackOctopus23
174 posts
#15 • 1 Y
Y by linjiah
The Unit Circle is also vital in trigonometry and in understanding the functions. This video helped me understand it a lot! Click to reveal hidden text. The unit circle is basically a circle of radius one. Remember that $cos$ is the $x$ and $sin$ is the $y$ if we are viewing it in the perspective of a graph.
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aok
352 posts
#16
Y by
Using unit circle as stated.
aok wrote:
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$
This post has been edited 1 time. Last edited by aok, May 16, 2025, 12:42 AM
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Shan3t
402 posts
#17
Y by
might be a bit advanced but Ceva's Theorem, and Extended LoS
This post has been edited 1 time. Last edited by Shan3t, May 16, 2025, 1:03 AM
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Shan3t
402 posts
#18
Y by
Shan3t wrote:
might be a bit advanced but Ceva's Theorem, and Extended LoS

also SAS(for area, side angle side), and Ceva's branches off to Menelaus's
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