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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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0 replies
jlacosta
May 1, 2025
0 replies
Evan Chen Multivariable Calculus Book
Existing_Human1   1
N 2 hours ago by MathCosine
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
1 reply
Existing_Human1
3 hours ago
MathCosine
2 hours ago
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
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
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
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
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
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
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
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
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
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
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
[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
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
Sequence, limit and number theory
KAME06   3
N Apr 5, 2025 by Rainbow1971
Source: Ecuador National Olympiad OMEC level U 2023 P6 Day 2
A positive integers sequence is defined such that, for all $n \ge 2$, $a_{n+1}$ is the greatest prime divisor of $a_1+a_2+...+a_n$. Find:
$$\lim_{n \rightarrow \infty} \frac{a_n}{n}$$
3 replies
KAME06
Feb 6, 2025
Rainbow1971
Apr 5, 2025
Sequence, limit and number theory
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Source: Ecuador National Olympiad OMEC level U 2023 P6 Day 2
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KAME06
158 posts
#1 • 1 Y
Y by Rainbow1971
A positive integers sequence is defined such that, for all $n \ge 2$, $a_{n+1}$ is the greatest prime divisor of $a_1+a_2+...+a_n$. Find:
$$\lim_{n \rightarrow \infty} \frac{a_n}{n}$$
This post has been edited 1 time. Last edited by KAME06, Feb 6, 2025, 8:33 PM
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Rainbow1971
35 posts
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And what are the definitions of $a_1$ and $a_2$?
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KAME06
158 posts
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Rainbow1971 wrote:
And what are the definitions of $a_1$ and $a_2$?

We just know they are two positive integers
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Rainbow1971
35 posts
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This is indeed a charming little problem. As it is a little convoluted in character, I wish to make it a little more straightforward by setting $a_1 = a_2 = 1$. The general setting with arbitrary starting values leads to "essentially" the same problem, but restricting those values to 1 helps to avoid unnecessary variables.

For convenience, I set $s_n = a_1 + a_2 + \ldots + a_{n-1}$, so that $a_n$ will be the biggest prime divisor of $s_n$. In order to gain some familiarity with the situation, the following table provides the relevant values for $n \in \{3, 4, \ldots, 30\}$:

$n$ $\quad$ $s_n$ $\quad$ $a_n$ $\quad$ $a_n/n$
3 $\quad \ $ 2 $\quad \ $ 2 $\quad \ $ 2/3
4 $\quad \ $ 4 $\quad \ $ 2 $\quad \ $ 1/2
5 $\quad \ $ 6 $\quad \ $ 3 $\quad \ $ 3/5
6 $\quad \ $ 9 $\quad \ $ 3 $\quad \ $ 1/2
7 $\quad \ $ 12 $\quad \ $ 3 $\quad \ $ 3/7
8 $\quad \ $ 15 $\quad \ $ 5 $\quad \ $ 5/8
9 $\quad \ $ 20 $\quad \ $ 5 $\quad \ $ 5/9
10 $\quad \ $ 25 $\quad \ $ 5 $\quad \ $ 1/2
11 $\quad \ $ 30 $\quad \ $ 5 $\quad \ $ 5/11
12 $\quad \ $ 35 $\quad \ $ 7 $\quad \ $ 7/12
13 $\quad \ $ 42 $\quad \ $ 7 $\quad \ $ 7/13
14 $\quad \ $ 49 $\quad \ $ 7 $\quad \ $ 1/2
15 $\quad \ $ 56 $\quad \ $ 7 $\quad \ $ 7/15
16 $\quad \ $ 63 $\quad \ $ 7 $\quad \ $ 7/16
17 $\quad \ $ 70 $\quad \ $ 7 $\quad \ $ 7/17
18 $\quad \ $ 77 $\quad \ $ 11 $\quad \ $ 11/18
19 $\quad \ $ 88 $\quad \ $ 11 $\quad \ $ 11/19
20 $\quad \ $ 99 $\quad \ $ 11 $\quad \ $ 11/20
21 $\quad \ $ 110 $\quad \ $ 11 $\quad \ $ 11/21
22 $\quad \ $ 121 $\quad \ $ 11 $\quad \ $ 1/2
23 $\quad \ $ 132 $\quad \ $ 11 $\quad \ $ 11/23
24 $\quad \ $ 143 $\quad \ $ 13 $\quad \ $ 13/24
25 $\quad \ $ 156 $\quad \ $ 13 $\quad \ $ 13/25
26 $\quad \ $ 169 $\quad \ $ 13 $\quad \ $ 1/2
27 $\quad \ $ 182 $\quad \ $ 13 $\quad \ $ 13/27
28 $\quad \ $ 195 $\quad \ $ 13 $\quad \ $ 13/28
29 $\quad \ $ 208 $\quad \ $ 13 $\quad \ $ 13/29
30 $\quad \ $ 221 $\quad \ $ 17 $\quad \ $ 17/30


With respect to this table, we will refer to the first column as the index or line number, to the second column as the $s$-column, to the third column as the $a$-column, and to the respective entries as $s$-values and $a$-values.

The table suggests to some extent that the limit of $(a_n/n)$ is $\tfrac{1}{2}$, and we will now examine that hypothesis.

When we take a look at our table, we see that it consists of sections of constant values for $a_n$. In lines 12 to 17, for example, we consistently have the value 7 in the $a$-column. We will now investigate these sections a little closer, focussing on the values of $s_n$ and $a_n$. For that purpose, we define $p_i$ to be the $i$-th prime number (in their natural increasing order), i.e. $p_1 = 2$, $p_2 = 3$ etc.

We start our investigation in line 5 which marks the beginning of the section of the value 3 for $a_n$. We observe that the $s$-value and the $a$-value, that is 6 and 3, can be written as $p_{i-1} \cdot p_i$ and $p_i$ for $i=2$. Plainly speaking, the $s$-value is the product of the corresponding prime in the $a$-column and the previous prime. We will show that this is no coincidence for the first line of such a section.

We make a sketch of an inductive argument: By inspection, we see that, at the beginning of the section with the $a$-value 3, we do indeed have $p_{i-1} \cdot p_i$ and $p_i$ in those two columns. By definition of $s_n$, the value in the $s$-column in the next line is $p_{i-1} \cdot p_i + p_i$ which is the same as $p_i \cdot (p_{i-1} + 1)$. If the value in the $a$-column of that line does not change, $p_i$ is added to the $s$-value in the following line once again, resulting in the value $p_i \cdot (p_{i-1} + 2)$ there, and as long as nothing changes in the $a$-column, the values in the $s$-column will be of the form $p_i \cdot (p_{i-1} + k)$, $k \in \{1, 2, 3, \ldots\}$ in the following lines.

The value in the $a$-column will change once we reach the smallest $k$ such that $p_i \cdot (p_{i-1} + k)$ has a prime factor $p$ larger than $p_i$. As two different prime numbers are always relatively prime, this is equivalent to the fact that $p$ divides $p_{i-1} + k$. We start with $k=1$, when $p_{i-1} + k$ is smaller than $p_i$ and also smaller than any candidate prime number $p$ (which must even be greater than $p_i$). Clearly, the first $k$ such that $p_{i-1} + k$ has a prime divisor greater than $p_i$ is the one with $p_{i-1} + k = p_{i+1}$, so that our new prime number will be $p = p_{i+1}$, which then does not only divide $p_{i-1} + k$, but will be equal to it.

This shows that the length of the section under investigation is $p_{i+1}-p_{i-1}$ lines (as that was the crucial value of $k$ which initiated a change in the $a$-column). The last line of that section will have the value $p_i \cdot (p_{i+1}-1)$ in the $s$-column and $p_i$ in the $a$-column, and the new section will therefore begin with a line that has $p_i \cdot p_{i+1}$ in the $s$-column and $p_{i+1}$ in the $a$-column. In particular, this shows that the values in the $a$-column run through all the prime numbers in a monotonously increasing way.

So far, we have described the values in the $a$-column in terms of the sequence $(p_n)$. Now we have to consider them as actual elements of the sequence $(a_n)$, which, loosely speaking, means that we have to find a relation between those values and the line number.

The crucial insight of our work so far is now that, in the $a$-column, the prime number $p_i$ prevails for exactly $p_{i+1}-p_{i-1}$ lines. Thus the prime number $p_2= 3$, which appears for the first time in line 5, is succeeded by $p_3= 5$ in line $5 + p_3 - p_1$. By induction, the prime number $p_i$ (for some integer $i$) will appear in the $a$-column for the first time in line
$$5 + (p_3 - p_1) + (p_4 - p_2) + (p_5 - p_3) + \ldots + (p_{i-1} - p_{i-3}) + (p_i - p_{i-2}),$$and this telescoping sum is the same as
$$5 + p_i + p_{i-1} - p_2 - p_1 = 5 + p_i + p_{i-1} - 3 - 2 = p_i + p_{i-1}.$$
This means nothing less than $$a_{p_i + p_{i-1}} = p_i,$$
and therefore $$\frac{a_n}{n} = \frac{p_i}{p_i + p_{i-1}} \quad \text{for $n = p_i + p_{i-1}$}.$$
As the $a$-value does not change within a section (by definition of a section), we can conclude that, at the end of the section, which comes $p_{i+1}-p_{i-1}-1$ lines later, we have $$\frac{a_n}{n} = \frac{p_i}{p_i+p_{i+1}-1} \quad \text{for $n = p_i + p_{i+1}-1$}.$$
Within a section, the values of $\tfrac{a_n}{n}$ are clearly strictly decreasing, as it is only the index $n$ which changes. Therefore, to establish the limit of $\tfrac{a_n}{n}$, which is the ultimate objective of this text, it suffices to focus on the values of $\tfrac{a_n}{n}$ at the beginning and at the end of each section. If the values of the subsequence at the beginning, i.e.
$$(\frac{p_i}{p_i + p_{i-1}}),$$and at the end, i.e.
$$(\frac{p_i}{p_i+p_{i+1}-1})$$converge to the same limit, the entire sequence $(\tfrac{a_n}{n})$ will converge to that same limit by the squeezing theorem. There is the (still open) conjecture that there are infinitely many twin primes. If we assume that the conjecture is true, this would easily show that the only possible limit of our two subsequences from above, and therefore the whole sequence, is indeed $\tfrac{1}{2}$.

To actually prove that limit statement, some sophisticated approximation of $p_i$ is needed. I am somewhat hesitant to proceed here, however, as I feel that this is beyond what is reasonable for a problem from a Math olympiad. To me, the attraction of this problem lies in uncovering the more elementary results from above. If others can produce an elementary proof of the limit statement, though, I am very interested in hearing about it.
This post has been edited 3 times. Last edited by Rainbow1971, Apr 6, 2025, 1:03 PM
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