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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
THREE People Meet at the SAME. TIME.
LilKirb   1
N 12 minutes ago by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
1 reply
LilKirb
4 hours ago
hellohi321
12 minutes ago
Computer too strong
Eyed   62
N 24 minutes ago by AR17296174
Source: 2020 ISL G6
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
62 replies
Eyed
Jul 20, 2021
AR17296174
24 minutes ago
Problem 5
blug   0
30 minutes ago
Source: Czech-Polish-Slovak Junior Match 2025 Problem 5
For every integer $n\geq 1$ prove that
$$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$
0 replies
blug
30 minutes ago
0 replies
Problem 1
blug   1
N 31 minutes ago by Primeniyazidayi
Source: Czech-Polish-Slovak Junior Match 2025 Problem 1
Find all primes $p, q, r$ such that
$$p^3+p^2+p+1=qr.$$
1 reply
blug
43 minutes ago
Primeniyazidayi
31 minutes ago
Problem 4
blug   0
34 minutes ago
Source: Czech-Polish-Slovak Junior Match 2025 Problem 4
Three non-negative integers are written on the board. In every step, the three numbers $(a, b, c)$ are being replaced with $a+b, b+c, c+a$. Find the biggest number of steps, after which the number $111$ will appear on the board.
0 replies
blug
34 minutes ago
0 replies
Problem 3
blug   0
37 minutes ago
Source: Czech-Polish-Slovak Junior Match 2025 Problem 3
In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that
$$AE+BD=KL.$$
0 replies
blug
37 minutes ago
0 replies
Problem 2
blug   0
39 minutes ago
Source: Czech-Polish-Slovak Junior Match 2025 Problem 2
Find all triangles that can be divided into congruent right-angled isosceles triangles with side lengths $1, 1, \sqrt{2}$.
0 replies
blug
39 minutes ago
0 replies
IMO Shortlist 2013, Geometry #2
lyukhson   79
N 40 minutes ago by CrazyInMath
Source: IMO Shortlist 2013, Geometry #2
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
79 replies
lyukhson
Jul 9, 2014
CrazyInMath
40 minutes ago
Goofy geometry
giangtruong13   0
an hour ago
Source: A Specialized School's Math Entrance Exam
Given the circle $(O)$, from $A$ outside the circle, draw tangents $AE,AF$ ($E,F$ are tangential points) and secant $ABC$ ($B,C$ lie on circle $O$, $B$ is between $A$ and $C$). $OA$ intersects $EF$ at $H$; $I$ is midpoint of $BC$. The line crossing $I$, paralleling with $CE$, intersects $EF$ at $D$. $CD$ intersects $AE$ at $K$. Let $N$ lie inside the triangle $FBC$ such that: $AF$=$AN$. From $N$ draw chords $BQ$, $RC$, $FP$ on circle $(O)$. Prove that: $PRQ$ is a isosceles triangle
0 replies
giangtruong13
an hour ago
0 replies
(a,b,c,d) of positive integers with 0<a,b,c,d <p-1 satisfy ad = bc mod p
parmenides51   4
N an hour ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1992 OMM P2
Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?
4 replies
parmenides51
Jul 29, 2018
FrancoGiosefAG
an hour ago
Collinearity with orthocenter
liberator   182
N an hour ago by CrazyInMath
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
182 replies
liberator
Jan 4, 2016
CrazyInMath
an hour ago
Inequalities
sqing   6
N 2 hours ago by sqing
Let $ a,b>0   $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1}  \leq  \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1}  \leq  \frac{2}{5} $$
6 replies
1 viewing
sqing
May 13, 2025
sqing
2 hours ago
Vieta's Formula.
BlackOctopus23   6
N 3 hours ago by Shan3t
Can someone help me understand Vieta's Formula? I am currently learning it for my class. I learned that for a polynomial of degree $n$, all the roots added will give $-\frac{a_{n-1}}{a_n}$. I also learned that if every single root, multiplies every single root, it will give $\frac{a_{n-2}}{a_n}$. I also learned that if all the roots are multiplied, it will give $-\frac{a_0}{a_n}$. Is this right? And is there any purpose for these equations?
6 replies
BlackOctopus23
Yesterday at 11:10 PM
Shan3t
3 hours ago
Number of elements in Set
girishpimoli   1
N 4 hours ago by alexheinis
Let $A=\left\{1,2,3,4,5,6,7\right\}$ and $B=\left\{3,6,7,9\right\}.$ Then the number of elements in the set ${C⊆A:C∩B=ϕ}$ is
1 reply
girishpimoli
4 hours ago
alexheinis
4 hours ago
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
350 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
350 posts
#11 • 1 Y
Y by linjiah
that is correct, cos x = a has 2 solutions (generally)
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aok
350 posts
#12 • 1 Y
Y by linjiah
for x btw
Z K Y
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aok
350 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
350 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
149 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
350 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
397 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
397 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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