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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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9 What is the most important topic in maths competition?
AVIKRIS   8
N 28 minutes ago by jb2015007
I think arithmetic is the most the most important topic in math competitions.
8 replies
AVIKRIS
an hour ago
jb2015007
28 minutes ago
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Preparing for Higher AIME+
PhoenixMathClub   5
N an hour ago by Andyluo
Hello, I am going to be a 7th grader next year and I really want to qualify for USAJMO in 8th grade, so far I have these goals reached

1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3 :(

This year on AIME something happened and I got a 3 :( on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
5 replies
PhoenixMathClub
5 hours ago
Andyluo
an hour ago
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Website to learn math
hawa   34
N 2 hours ago by iwastedmyusername
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
34 replies
1 viewing
hawa
Apr 9, 2025
iwastedmyusername
2 hours ago
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2500th post
Solocraftsolo   19
N 2 hours ago by b2025tyx
i keep forgetting to do these...


2500 is cool.

i am not very sentimental so im not going to post a math story or anything.

here are some problems though

p1p2p3

p4
19 replies
Solocraftsolo
Apr 16, 2025
b2025tyx
2 hours ago
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Calculus BC help
needcalculusasap45   0
5 hours ago
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

0 replies
needcalculusasap45
5 hours ago
0 replies
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Inequalities
sqing   9
N 5 hours ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
9 replies
sqing
Apr 4, 2025
sqing
5 hours ago
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Geometry
German_bread   1
N Today at 12:01 PM by vanstraelen
A semicircle k with radius r is constructed over the line segment ST. Let D be a point on the line segment ST that is different from S and T. The two squares ABCD and DEF G lie in the half-plane of the semicircle such that points B and F lie on the semicircle k and points S, C, D, E, and T lie on a straight line in that order. (Points A and/or G can also lie outside the semicircle if necessary.)
Investigate whether the sum of the areas of the squares ABCD and DEFG depends on the position of point D on the line segment ST.

German math olympiad, class 9, 2022
1 reply
German_bread
Today at 10:00 AM
vanstraelen
Today at 12:01 PM
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Indonesia Regional MO 2019 Part A
parmenides51   22
N Today at 10:43 AM by SomeonecoolLovesMaths
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
22 replies
parmenides51
Nov 11, 2021
SomeonecoolLovesMaths
Today at 10:43 AM
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Maximizing the Sum of Minimum Differences in Permutations
chinawgp   0
Today at 10:20 AM
Problem Statement

Given a positive integer n \geq 3 , consider a permutation \pi = (a_1, a_2, \dots, a_n) of \{1, 2, \dots, n\} . For each i ( 1 \leq i \leq n-1 ), define d_i as the minimum absolute difference between a_i and any subsequent element a_j ( j > i ), i.e.,
d_i = \min \{ |a_i - a_j| \mid j > i \}.

Let S_n denote the maximum possible sum of d_i over all permutations of \{1, \dots, n\} , i.e.,
S_n = \max_{\pi} \sum_{i=1}^{n-1} d_i.

Proposed Construction

I found a method to construct a permutation that seems to maximize \sum d_i :
1. Fix a_{n-1} = 1 and a_n = n .
2. For each i (from n-2 down to 1 ):
- Sort a_{i+1}, a_{i+2}, \dots, a_n in increasing order.
- Compute the gaps between consecutive elements.
- Place a_i in the middle of the largest gap (if the gap has even length, choose the smaller midpoint).

Partial Results

1. I can prove that 1 and n must occupy the last two positions. Otherwise, moving either 1 or n further right does not decrease \sum d_i .
2. The construction greedily maximizes each d_i locally, but I’m unsure if this ensures global optimality.

Request for Help

- Does this construction always yield the maximum S_n ?
- If yes, how can we rigorously prove it? (Induction? Exchange arguments?)
- If no, what is the correct approach?

Observations:
- The construction works for small n (e.g., n=3,4,5,...,12 ).
- The problem resembles optimizing "minimum gaps" in permutations.

Any insights or references would be greatly appreciated!
0 replies
chinawgp
Today at 10:20 AM
0 replies
J
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
parmenides51   9
N Today at 9:11 AM by AshAuktober
Find the number of integer solutions to $||x| - 2020| < 5$.
9 replies
parmenides51
Jan 18, 2021
AshAuktober
Today at 9:11 AM
J
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Geometry
German_bread   2
N Today at 8:31 AM by German_bread
Let P be a point in a square ABCD. The lengths of segments PA, PB, PC are 17, 11 and 5 respectively. Determine the area of the square and if it can’t be determined exactly, all possible values are to be listed.

German math Olympiad, Class 9, 2024

It’s my first time posting - please excuse any mistakes
2 replies
German_bread
Yesterday at 7:59 PM
German_bread
Today at 8:31 AM
J
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A Loggy Problem from Pythagoras
Mathzeus1024   6
N Today at 8:01 AM by Mathzeus1024
Prove or disprove: $\exists x \in \mathbb{R}^{+}$ such that $\ln(x), \ln(2x), \ln(3x)$ are the lengths of a right triangle.
6 replies
Mathzeus1024
Yesterday at 10:55 AM
Mathzeus1024
Today at 8:01 AM
J
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Nesbitt inequality
Mathskidd   1
N Today at 7:20 AM by sqing


$$ $$Would anyone tell me whether the number of ways for proving Nesbitt inequality more than one hundred ?
1 reply
Mathskidd
Today at 5:08 AM
sqing
Today at 7:20 AM
J
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Algebra Problems
ilikemath247365   10
N Today at 4:25 AM by lgx57
Find all real $(a, b)$ with $a + b = 1$ such that

$(a + \frac{1}{a})^{2} + (b + \frac{1}{b})^{2} = \frac{25}{2}$.
10 replies
ilikemath247365
Apr 14, 2025
lgx57
Today at 4:25 AM
J
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J
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An algebra math problem
AVY2024   6
N Yesterday at 6:03 PM by Roger.Moore
Solve for a,b
ax-2b=5bx-3a
6 replies
AVY2024
Apr 8, 2025
Roger.Moore
Yesterday at 6:03 PM
J
An algebra math problem
G H J
Reveal topic tags
algebra
nmtc
equation
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G H BBookmark kLocked kLocked NReply
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AVY2024
22 posts
AVY2024
#1 h  Apr 8, 2025, 10:55 AM
Y by
Solve for a,b
ax-2b=5bx-3a
Z K Y
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Inaaya
269 posts
Inaaya
#2 h  Apr 8, 2025, 1:40 PM • 1 Y
Y by Exponent11
There are 3 variables and one equation, this thing can theoretically have infinite solution sets
Z K Y
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sp0rtman00000
2 posts
sp0rtman00000
#3 h  Apr 11, 2025, 11:03 AM
Y by
On the R or on the N?
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Pengu14
505 posts
Pengu14
#5 h  Apr 11, 2025, 3:24 PM
Y by
Inaaya wrote:
There are 3 variables and one equation, this thing can theoretically have infinite solution sets

I believe they’re looking for a and b such that the equation holds for ALL x.
Z K Y
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Pengu14
505 posts
Pengu14
#6 h  Apr 11, 2025, 3:25 PM
Y by
AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.
Z K Y
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Runner1600
12 posts
Runner1600
#7 h  Apr 11, 2025, 3:56 PM
Y by
Pengu14 wrote:
AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.

That is what I got
Z K Y
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Roger.Moore
4 posts
Roger.Moore
#8 h  Yesterday at 6:03 PM
Y by
The x there are parameter?
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