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.

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0 replies
+1 w
jwelsh
Jul 1, 2025
0 replies
a cute combinatorics (?) problem
pzzd   4
N 43 minutes ago by maromex
here’s a cute little problem that can be solved with combinatorics, but is also related to some very common sequences in mathematics :3

say you have a $2$-inch-wide rectangle of some length $l$, and a bunch of $2$x$1$ dominos. how many different ways can you completely cover the rectangle with dominos? you can place the dominos horizontally or vertically - for example, for a $2$-by-$3$ rectangle, a valid arrangement of dominos is $1$ vertical domino on the left and $2$ horizontal dominos on the right.

hope you find this interesting!
4 replies
pzzd
Today at 2:48 PM
maromex
43 minutes ago
Trigonometry equation practice
ehz2701   6
N 2 hours ago by vanstraelen
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

Leaderboard

problem set 1a

problem set 2a

problem set 2b
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

[center] $\frac{1}{2} = \cos 36 - \sin 18$. [/center]

Many of these seem to be reducible to this. The half can be written as $\cos 60 = \sin 30$, and $\cos 36 = \sin 54$, $\sin 18 = \cos 72$. This is proven in solution 1a-1. We will refer to this as Trick 1.
6 replies
ehz2701
Yesterday at 8:48 AM
vanstraelen
2 hours ago
Easy geometry problem
menseggerofgod   5
N 4 hours ago by ehz2701
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
5 replies
menseggerofgod
Today at 2:47 AM
ehz2701
4 hours ago
Geometry easy
AlexCenteno2007   4
N 4 hours ago by AlexCenteno2007
In triangle ABC, if angle B=120°, AB=5u and BC=15u. Draw the interior bisector BE. Calculate BE
4 replies
AlexCenteno2007
Friday at 10:55 PM
AlexCenteno2007
4 hours ago
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
Putnam 2014 A5
Kent Merryfield   18
N Today at 9:04 AM by Assassino9931
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$
18 replies
Kent Merryfield
Dec 7, 2014
Assassino9931
Today at 9:04 AM
Original Problem: Geometry and Functions
wonderboy807   2
N May 25, 2025 by LilKirb
For any positive integer n, let F(n) be the number of interior diagonals in a convex polygon with n+3 sides. Find 1/(F(1)) + 1/(F(2)) + ... + 1/F(10))

Answer: Click to reveal hidden text
2 replies
wonderboy807
May 24, 2025
LilKirb
May 25, 2025
Original Problem: Geometry and Functions
G H J
G H BBookmark kLocked kLocked NReply
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wonderboy807
29 posts
#1
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For any positive integer n, let F(n) be the number of interior diagonals in a convex polygon with n+3 sides. Find 1/(F(1)) + 1/(F(2)) + ... + 1/F(10))

Answer: Click to reveal hidden text
Z K Y
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wonderboy807
29 posts
#2
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Solution: Click to reveal hidden text
This post has been edited 1 time. Last edited by wonderboy807, May 25, 2025, 12:08 AM
Reason: I forgot a period.
Z K Y
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LilKirb
63 posts
#3
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Cool problem
Solution
Z K Y
N Quick Reply
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