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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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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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Frankenstein FE
NamelyOrange   3
N 42 minutes ago by jasperE3
[quote = My own problem]Solve the FE $f(x)+f(-x)=2f(x^2)$ over $\mathbb{R}$. Ignore "pathological" solutions.[/quote]

How do I solve this? I made this while messing around, and I have no clue as to what to do...
3 replies
NamelyOrange
Jul 19, 2024
jasperE3
42 minutes ago
J
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Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
Vulch   1
N 2 hours ago by aidan0626
Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
1 reply
Vulch
2 hours ago
aidan0626
2 hours ago
J
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Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
Vulch   1
N 2 hours ago by aidan0626
Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
1 reply
Vulch
3 hours ago
aidan0626
2 hours ago
J
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   0
2 hours ago
Find the domain and range of $f(x)=2-|x-5|.$
0 replies
1 viewing
Vulch
2 hours ago
0 replies
J
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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
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Range of 2 parameters and Convergency of Improper Integral
Kunihiko_Chikaya   2
N Yesterday at 9:37 PM by Hello_Kitty
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.
2 replies
Kunihiko_Chikaya
Aug 21, 2012
Hello_Kitty
Yesterday at 9:37 PM
J
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Does the sequence log(1+sink)/k converge?
tom-nowy   2
N Yesterday at 9:25 PM by Hello_Kitty
Source: Question arising while viewing https://artofproblemsolving.com/community/c7h3556569
Does the sequence $$ \frac{\ln(1+\sin k)}{k} \;\;\;(k=1,2,3,\ldots) $$converge?
2 replies
tom-nowy
Yesterday at 10:35 AM
Hello_Kitty
Yesterday at 9:25 PM
J
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integration
We2592   2
N Yesterday at 7:53 PM by Litvinov
Q) solve the integration
$\int_{0}^{\alpha} \frac{d\theta}{\sqrt{cos\theta-cos\alpha}}$
2 replies
We2592
Apr 25, 2025
Litvinov
Yesterday at 7:53 PM
J
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Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$
Vulch   4
N Yesterday at 5:35 PM by Vulch
Respected users,
I am asking for better solution of the following problem with excellent explanation.
Thank you!

Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$
4 replies
Vulch
Yesterday at 2:33 AM
Vulch
Yesterday at 5:35 PM
J
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power of matrix
teomihai   3
N Yesterday at 4:31 PM by teomihai
Find $A^{n}$ ,where $A=\begin{pmatrix}1&-\frac{1}{5}-\frac{2}{5}i\\2+i&-i&\end{pmatrix}$
we now $i^2=-1$,and $n$ it is positiv integer number.
3 replies
teomihai
Yesterday at 12:15 PM
teomihai
Yesterday at 4:31 PM
J
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Putnam 1952 B1
centslordm   3
N Yesterday at 4:24 PM by Ianis
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
3 replies
centslordm
May 30, 2022
Ianis
Yesterday at 4:24 PM
J
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Putnam 1952 A1
centslordm   4
N Yesterday at 3:21 PM by centslordm
Let \[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\]be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.
4 replies
centslordm
May 29, 2022
centslordm
Yesterday at 3:21 PM
J
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Putnam 1951 B3
centslordm   4
N Yesterday at 3:18 PM by centslordm
Show that if $x$ is positive, then \[ \log_e (1 + 1/x) > 1 / (1 + x).\]
4 replies
centslordm
May 25, 2022
centslordm
Yesterday at 3:18 PM
J
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Putnam 1980 B3
sqrtX   2
N Yesterday at 2:10 PM by Etkan
Source: Putnam 1980
For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$
2 replies
sqrtX
Apr 1, 2022
Etkan
Yesterday at 2:10 PM
J
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J
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AB <2CD wanted, tetrahedron ABCD,<ACB=<ADB=90^o,AC=CD=DB (2015 Polish JMO R1 p7)
parmenides51   0
Apr 5, 2021
A tetrahedron $ABCD$ is given where $\angle ACB =\angle ADB = 90^o$ and $AC = CD = DB$. Prove that $AB <2 CD$.
0 replies
parmenides51
Apr 5, 2021
0 replies
J
AB <2CD wanted, tetrahedron ABCD,<ACB=<ADB=90^o,AC=CD=DB (2015 Polish JMO R1 p7)
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parmenides51
30650 posts
parmenides51
#1 h  Apr 5, 2021, 8:30 PM • 1 Y
Y by samrocksnature
A tetrahedron $ABCD$ is given where $\angle ACB =\angle ADB = 90^o$ and $AC = CD = DB$. Prove that $AB <2 CD$.
This post has been edited 1 time. Last edited by parmenides51, Apr 5, 2021, 8:34 PM
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