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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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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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
Cool vieta sum
Kempu33334   6
N 2 minutes ago by Lankou
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
6 replies
Kempu33334
Yesterday at 11:44 PM
Lankou
2 minutes ago
đề hsg toán
akquysimpgenyabikho   3
N 41 minutes ago by Lankou
làm ơn giúp tôi giải đề hsg

3 replies
akquysimpgenyabikho
Apr 27, 2025
Lankou
41 minutes ago
A problem with a rectangle
Raul_S_Baz   13
N 2 hours ago by undefined-NaN
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
13 replies
Raul_S_Baz
Apr 26, 2025
undefined-NaN
2 hours ago
Putnam 1958 February A5
sqrtX   4
N 4 hours ago by Safal
Source: Putnam 1958 February
Show that the integral equation
$$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$
4 replies
sqrtX
Jul 18, 2022
Safal
4 hours ago
Miklós Schweitzer 1956- Problem 1
Coulbert   1
N 5 hours ago by NODIRKHON_UZ
1. Solve without use of determinants the following system of linear equations:

$\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$),

where $\alpha$ is a fixed real number. (A. 7)
1 reply
Coulbert
Oct 9, 2015
NODIRKHON_UZ
5 hours ago
D1021 : Does this series converge?
Dattier   3
N 5 hours ago by Dattier
Source: les dattes à Dattier
Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?
3 replies
Dattier
Apr 26, 2025
Dattier
5 hours ago
If a matrix exponential is identity, does it follow the initial matrix is zero?
bakkune   5
N 6 hours ago by loup blanc
This might be a really dumb question, but I have neither a rigorous proof nor a counter example.

For any square matrix $\mathbf{A}$, define
$$
e^{\mathbf{A}} = \mathbf{I} + \sum_{n=1}^{+\infty} \frac{1}{n!}\mathbf{A}^n
$$where $\mathbf{I}$ is the identity matrix. If for some matrix $\mathbf{A}$ that $e^{\mathbf{A}}$ is identity, does it follow that $\mathbf{A}$ is zero?
5 replies
bakkune
Mar 4, 2025
loup blanc
6 hours ago
Find the domain and range of $f(x)=2-|x-5|.$
Vulch   1
N Today at 12:13 PM by Mathzeus1024
Find the domain and range of $f(x)=2-|x-5|.$
1 reply
Vulch
Today at 2:07 AM
Mathzeus1024
Today at 12:13 PM
Range of 2 parameters and Convergency of Improper Integral
Kunihiko_Chikaya   3
N Today at 11:37 AM by Mathzeus1024
Source: 2012 Kyoto University Master Course in Mathematics
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
3 replies
Kunihiko_Chikaya
Aug 21, 2012
Mathzeus1024
Today at 11:37 AM
Matrix Row and column relation.
Schro   6
N Today at 6:20 AM by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
Today at 6:20 AM
A small problem in group theory
qingshushuxue   2
N Today at 4:42 AM by qingshushuxue
Assume that $G,A,B,C$ are group. If $G=\left( AB \right) \bigcup \left( CA \right)$, prove that $G=AB$ or $G=CA$.

where $$A,B,C\subset G,AB\triangleq \left\{ ab:a\in A,b\in B \right\}.$$
2 replies
qingshushuxue
Today at 2:06 AM
qingshushuxue
Today at 4:42 AM
Putnam 1958 February A4
sqrtX   2
N Today at 2:14 AM by centslordm
Source: Putnam 1958 February
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$whenever the denominator of the left-hand side is different from $0$.
2 replies
sqrtX
Jul 18, 2022
centslordm
Today at 2:14 AM
analysis
Hello_Kitty   2
N Yesterday at 10:37 PM by Hello_Kitty
what is the range of $f=x+2y+3z$ for any positive reals satifying $z+2y+3x<1$ ?
2 replies
Hello_Kitty
Yesterday at 9:59 PM
Hello_Kitty
Yesterday at 10:37 PM
Putnam 1958 February A1
sqrtX   2
N Yesterday at 10:32 PM by centslordm
Source: Putnam 1958 February
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.
2 replies
sqrtX
Jul 18, 2022
centslordm
Yesterday at 10:32 PM
trigonometric functions
VivaanKam   9
N Yesterday at 7:12 PM by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Tuesday at 8:29 PM
aok
Yesterday at 7:12 PM
trigonometric functions
G H J
G H BBookmark kLocked kLocked NReply
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VivaanKam
155 posts
#1
Y by
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
Z K Y
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Lijin
222 posts
#2
Y by
Are you talking about graphing them or just the basic ratios?
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Yiyj1
1265 posts
#4
Y by
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.
Z K Y
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aok
330 posts
#5
Y by
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, Tuesday at 10:48 PM
Z K Y
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VivaanKam
155 posts
#6
Y by
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]
Z K Y
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VivaanKam
155 posts
#7
Y by
That’s cool! So if you have the lengths of a triangle you can find its angles?
Z K Y
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VivaanKam
155 posts
#8
Y by
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 ?
Z K Y
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VivaanKam
155 posts
#9
Y by
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
2980 posts
#10
Y by
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, Yesterday at 6:39 PM
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aok
330 posts
#11
Y by
that is correct, cos x = a has 2 solutions (generally)
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