Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
Original Question #8
Siopao_Enjoyer   1
N 9 minutes ago by Siopao_Enjoyer
Let $x$, $y$, $z$ be real numbers such that:
\[
    \begin{cases}
        x+y+z&=1\\
        x^2+y^2+z^2&=12 \\
        x^3+y^3+z^3&=18
    \end{cases}
    \]Find the value of $xyz$.

Answer Confirmation
1 reply
Siopao_Enjoyer
12 minutes ago
Siopao_Enjoyer
9 minutes ago
Easy geometry problem
menseggerofgod   6
N 11 minutes ago by menseggerofgod
ABC is a right triangle, right at B, in which the height BD is drawn. E is a point on side BC such that AE = EC = 8. If BD is 6 and DE = k , find k
6 replies
menseggerofgod
Today at 2:47 AM
menseggerofgod
11 minutes ago
Amc10 question
littyyin   0
13 minutes ago
What's the best way to prepare for amc10 to qualify for AIME? I've been grinding alcumus on insane difficulty and Mathcounts trainer but when I look at past AMC10 exams, such as the 2022 Amc10, I find myself struggling to answer 15-20.
0 replies
littyyin
13 minutes ago
0 replies
Digits problem
menseggerofgod   0
18 minutes ago
Jean came up with a positive integer that is divisible by 13, whose digits are not zero and distinct two by two. He noticed that in this number two digits can be interchanged so that the result is also divisible by 13. What is the smallest number of digits that Jean's number could have?
a)14 b)5 c)6 d)7 e)8
0 replies
menseggerofgod
18 minutes ago
0 replies
Putnam 2003 B3
btilm305   35
N Today at 2:50 PM by SomeonecoolLovesMaths
Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\](Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)
35 replies
btilm305
Jun 23, 2011
SomeonecoolLovesMaths
Today at 2:50 PM
Simple integuration
obihs   1
N Today at 2:21 PM by Litvinov
Source: Own
Find the value of
$$\int_1^2\dfrac{\ln x}{(x^2-2x+2)^2}dx$$
1 reply
obihs
Today at 8:44 AM
Litvinov
Today at 2:21 PM
Estimating the Density
zqy648   0
Today at 1:11 PM
Source: 2024 May 谜之竞赛-6
Given non-empty subset \( I \) of the set of positive integers, a positive integer \( n \) is called good if for every prime factor \( p \) of \( n \), \( \nu_p(n) \in I \). For a positive real number \( x \), let \( S(x) \) denote the number of good numbers not exceeding \( x \).

Determine all positive real numbers \( C \) and \( \alpha \) such that \(
\lim\limits_{x \to +\infty} \dfrac{S(x)}{x^\alpha} = C.
\)

Proposed by Zhenqian Peng, High School Affiliated to Renmin University of China
0 replies
zqy648
Today at 1:11 PM
0 replies
Show that if \( d_3 < \frac{d_1}{3} \), then there exist two other positive diag
Martin.s   3
N Today at 12:26 PM by Martin.s
Let
\[
D = \begin{bmatrix}
d_1 & 0 & 0 \\
0 & d_2 & 0 \\
0 & 0 & d_3
\end{bmatrix}, \quad
T = \begin{bmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]where \( d_1, d_2, d_3 \) are positive and \( d_1 \ge d_3 \).

Show that if \( d_3 < \frac{d_1}{3} \), then there exist two other positive diagonal matrices \( D_1 \) and \( D_2 \) such that \( D, D_1, D_2 \) are distinct but \( DT, D_1T, D_2T \) have the same eigenvalues.

Show also that if \( d_3 > \frac{d_1}{3} \) and \( D_1 \) is a positive diagonal matrix distinct from \( D \), then \( DT \) and \( D_1T \) must have different eigenvalues.
3 replies
Martin.s
Jun 23, 2025
Martin.s
Today at 12:26 PM
Inequality
Martin.s   4
N Today at 12:25 PM by Martin.s


For \( n = 2, 3, \dots \), the following inequalities hold:

\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n},
\]
and

\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{1}{5}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \frac{\pi}{2}.
\]
4 replies
Martin.s
Jun 23, 2025
Martin.s
Today at 12:25 PM
Analytic Number Theory
zqy648   1
N Today at 11:59 AM by zqy648
Source: 2024 December 谜之竞赛-6
For positive integer \( n \), define
\[S_n = \{(a, b) \in \mathbb{N}_+^2 \mid a, b < \sqrt{n} \text{ and } n \mid a^2 + b^3 + 1\}. \]Prove that there exists a positive real number \(\varepsilon\) such that for all integers \(n \geq 10\), \(
\left| S_n \right| < n^{\frac{1}{2} - \frac{\varepsilon}{\ln \ln n}}.
\)

Proposed by Yuxing Ye
1 reply
zqy648
Today at 9:36 AM
zqy648
Today at 11:59 AM
infinite series
Martin.s   0
Today at 11:37 AM
Can the infinite series

$$\sum\limits_{n=1}^{\infty}\frac{x^n}{n!}\ln(n+\alpha)$$
be expressed in terms of known functions and constants?
0 replies
Martin.s
Today at 11:37 AM
0 replies
2023 Putnam A2
giginori   23
N Today at 10:05 AM by BlayzyMath
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2 n$; that is to say, $p(x)=$ $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$ for some real coefficients $a_0, \ldots, a_{2 n-1}$. Suppose that $p(1 / k)=k^2$ for all integers $k$ such that $1 \leq|k| \leq n$. Find all other real numbers $x$ for which $p(1 / x)=x^2$.
23 replies
giginori
Dec 3, 2023
BlayzyMath
Today at 10:05 AM
Something Proposer Said is Easy
EthanWYX2009   1
N Today at 9:59 AM by Martin.s
Source: 2023 September 谜之竞赛-7
For a positive integer \( n \), let \( P_n \) denote the product of all prime numbers not exceeding \( n \).

Prove that there exists a constant \( c > 0 \) such that for any sufficiently large integer \( n \), if all integers not exceeding \( P_n \) and coprime with \( P_n \) are arranged in a sequence, then there exist two adjacent numbers in this sequence whose difference is at least \( c \cdot \frac{n \cdot \ln n}{(\ln \ln n)^2} \).

Furthermore, consider whether this bound can be strengthened to \( c \cdot \frac{n \cdot \ln n \cdot \ln \ln \ln n}{(\ln \ln n)^2} \).

Created by Mucong Sun, Tianjin Experimental Binhai School
1 reply
EthanWYX2009
Yesterday at 10:57 AM
Martin.s
Today at 9:59 AM
Great Use of Weyl Distribution
zqy648   0
Today at 9:33 AM
Source: 2025 March 谜之竞赛-6
Prove that for any real number \(\varepsilon > 0\), there exists a positive integer \(N\) such that for any prime \(p > N\) and any primitive root \(g\) modulo \(p\), if we define the set
\[  
S = \left\{(i, j) \mid 1 \leq i < j \leq p - 1, 
\left\{ \frac{g^i}{p} \right\} > \left\{ \frac{g^j}{p} \right\} \right\}
\]where \(\{x\}\) denotes the fractional part of the real number \(x\), then
\[  
\left( \frac{1}{4} - \varepsilon \right) p^2 < |S| < \left( \frac{1}{4} + \varepsilon \right) p^2.  
\]Created by Zhenyu Dong and Chunji Wang
0 replies
zqy648
Today at 9:33 AM
0 replies
U2 Original Problem
NeoAzure   0
Jun 1, 2025
In the International Smithery Olympiad (ISO), a blacksmith must forge a lance consisting of a 4 inch long cylindrical wooden handle (with volume 25\pi inches) attached to a 12 inch long aluminum conical head whose tip (the outermost 10% of its length) is plated in steel. If the radius of the cylindrical handle is half the radius of the cone, how much steel plating is needed to cover the tip? Answer in units square inches.

Note: Pardon for the lack of formatting, my account is new and I can't use math mode. Better check out my solution in the pdf for LaTeX formatting and better organization.

Answer

Solution
0 replies
NeoAzure
Jun 1, 2025
0 replies
U2 Original Problem
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NeoAzure
21 posts
#1
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In the International Smithery Olympiad (ISO), a blacksmith must forge a lance consisting of a 4 inch long cylindrical wooden handle (with volume 25\pi inches) attached to a 12 inch long aluminum conical head whose tip (the outermost 10% of its length) is plated in steel. If the radius of the cylindrical handle is half the radius of the cone, how much steel plating is needed to cover the tip? Answer in units square inches.

Note: Pardon for the lack of formatting, my account is new and I can't use math mode. Better check out my solution in the pdf for LaTeX formatting and better organization.

Answer

Solution
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a