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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.

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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
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Evan Chen Multivariable Calculus Book
Existing_Human1   2
N a few seconds ago by RedChameleon
What do you guys think of Evan Chen's multivariable calculus book (or whatever you want to call it), through MIT? Is it useful for learning multivariable calculus? Here is a link: book
2 replies
Existing_Human1
4 hours ago
RedChameleon
a few seconds ago
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A Collection of Good Problems from my end
SomeonecoolLovesMaths   24
N 4 hours ago by ReticulatedPython
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
24 replies
SomeonecoolLovesMaths
May 4, 2025
ReticulatedPython
4 hours ago
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n and n+100 have odd number of divisors (1995 Belarus MO Category D P2)
jasperE3   4
N Yesterday at 9:50 PM by KTYC
Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors.
4 replies
jasperE3
Apr 6, 2021
KTYC
Yesterday at 9:50 PM
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Closed form expression of 0.123456789101112....
ReticulatedPython   3
N Yesterday at 8:15 PM by ReticulatedPython
Is there a closed form expression for the decimal number $$0.123456789101112131415161718192021...$$which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
3 replies
ReticulatedPython
Yesterday at 8:05 PM
ReticulatedPython
Yesterday at 8:15 PM
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primes and perfect squares
Bummer12345   5
N Yesterday at 8:08 PM by Shan3t
If $p$ and $q$ are primes, then can $2^p + 5^q + pq$ be a perfect square?
5 replies
Bummer12345
Monday at 5:08 PM
Shan3t
Yesterday at 8:08 PM
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trapezoid
Darealzolt   1
N Yesterday at 7:38 PM by vanstraelen
Let \(ABCD\) be a trapezoid such that \(A, B, C, D\) lie on a circle with center \(O\), and side \(AB\) is parallel to side \(CD\). Diagonals \(AC\) and \(BD\) intersect at point \(M\), and \(\angle AMD = 60^\circ\). It is given that \(MO = 10\). It is also known that the difference in length between \(AB\) and \(CD\) can be expressed in the form \(m\sqrt{n}\), where \(m\) and \(n\) are positive integers and \(n\) is square-free. Compute the value of \(m + n\).
1 reply
Darealzolt
Yesterday at 2:03 AM
vanstraelen
Yesterday at 7:38 PM
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Rolles theorem
sasu1ke   7
N Yesterday at 7:27 PM by GentlePanda24

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[ f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1. \]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

7 replies
sasu1ke
May 3, 2025
GentlePanda24
Yesterday at 7:27 PM
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Polynomial Minimization
ReticulatedPython   1
N Yesterday at 5:36 PM by clarkculus
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 $x=\frac{n}{n+1}.$
1 reply
ReticulatedPython
Yesterday at 5:07 PM
clarkculus
Yesterday at 5:36 PM
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Easy one
irregular22104   0
Yesterday at 5:03 PM
Given two positive integers a,b written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 2.5, then the numbers on the board after step 1 are 2,5,7; after step 2 are 2,5,7,9,12;...
1) With a = 3; b = 12, prove that the number 2024 cannot appear on the board.
2) With a = 2; b = 3, prove that the number 2024 can appear on the board.
0 replies
irregular22104
Yesterday at 5:03 PM
0 replies
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This shouldn't be a problem 15
derekli   2
N Yesterday at 4:09 PM by aarush.rachak11
Hey guys I was practicing AIME and came across this problem which is definitely misplaced. It asks for the surface area of a plane within a cylinder which we can easily find out using a projection that is easy to find. I think this should be placed in problem 10 or below. What do you guys think?
2 replies
derekli
Yesterday at 2:15 PM
aarush.rachak11
Yesterday at 4:09 PM
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Morphism in a ring makes it a field
RobertRogo   0
Yesterday at 4:02 PM
Source: Daniel Jinga, Ionel Popescu, RNMO SHL, 2003
Let $A$ be a ring with unity in which $1+1 \neq 0$ and there is a morphism $f$ from the group $(A,+)$ to the monoid $(A,\cdot)$ such that for all $a\in A\setminus \{0\}$, there is a $b \in A$ such that $f(b)=a^2$. Prove that $A$ is a field.
0 replies
RobertRogo
Yesterday at 4:02 PM
0 replies
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Regular tetrahedron
vanstraelen   7
N Yesterday at 3:46 PM by ReticulatedPython
Given the points $O(0,0,0),A(1,0,0),B(\frac{1}{2},\frac{\sqrt{3}}{2},0)$
a) Determine the point $C$, above the xy-plane, such that the pyramid $OABC$ is a regular tetrahedron.
b) Calculate the volume.
c) Calculate the radius of the inscribed sphere and the radius of the circumscribed sphere.
7 replies
vanstraelen
May 4, 2025
ReticulatedPython
Yesterday at 3:46 PM
J
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[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1
parmenides51   4
N Yesterday at 3:44 PM by Kizaruno
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?

(Here $[t]$ denotes the area of the geometrical figure$ t$.)
4 replies
parmenides51
Jan 18, 2021
Kizaruno
Yesterday at 3:44 PM
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Convergent series with weight becomes divergent
P_Fazioli   7
N Yesterday at 3:07 PM by P_Fazioli
Initially, my problem was : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ?

Thinking about the continuous case : if $g:\mathbb{R}_+\longrightarrow\mathbb{R}$ is continuous, positive with $g(x)\underset{{x}\longrightarrow{+\infty}}\longrightarrow +\infty$, does $f$ continuous and positive exist on $\mathbb{R}_+$ such that $\displaystyle\int_0^{+\infty}f(x)\text{d}x$ converges and $\displaystyle\int_0^{+\infty}f(x)g(x)\text{d}x$ diverges ?

To the last question, the answer seems to be yes if $g$ is in the $\mathcal{C}^1$ class, increasing : I chose $f=\dfrac{g'}{g^2}$. With this idea, I had the idea to define $a_n=\dfrac{b_{n+1}-b_n}{b_n^2}$ but it is not clear that it is ok, even if $(b_n)$ is increasing.

Now I have some questions !

1) The main problem : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ? And if $(b_n)$ is increasing ?
2) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\left(\frac{b_{n+1}}{b_n}-1\right)$ diverges ?
3) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ converges ?
4) if $(b_n)$ is positive increasing and such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}$ does not converge to $1$, can $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ diverge ?
5) for the continuous case, is it true if we suppose $g$ only to be continuous ?

7 replies
P_Fazioli
Monday at 5:37 AM
P_Fazioli
Yesterday at 3:07 PM
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determinant practice
Semcio   1
N Apr 16, 2025 by Saucepan_man02
let $A=(\varepsilon^{i \cdot j})_{ij}$ where$1\leq i,j \leq p$ and $\varepsilon=e^{\frac{2i\pi}{p}}$ is root of unity
calculate $\det(A)$
1 reply
Semcio
Jul 12, 2020
Saucepan_man02
Apr 16, 2025
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determinant practice
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Semcio
121 posts
Semcio
#1 h  Jul 12, 2020, 6:03 PM
Y by
let $A=(\varepsilon^{i \cdot j})_{ij}$ where$1\leq i,j \leq p$ and $\varepsilon=e^{\frac{2i\pi}{p}}$ is root of unity
calculate $\det(A)$
This post has been edited 2 times. Last edited by Semcio, Jul 13, 2020, 11:10 AM
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Saucepan_man02
1331 posts
Saucepan_man02
#2 h  Apr 16, 2025, 2:07 AM
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I got $\det(A) = (-1)^{n+1} \prod_{n \ge j>i \ge 1} (\epsilon^j-\epsilon^i)$
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