right triangle from BB,CA,DC if AD = BC, <ADB +<ACB=

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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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

J

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k i Adding contests to the Contest Collections

dcouchman 1

N Apr 5, 2023 by v_Enhance

Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add

1 reply

dcouchman

Sep 9, 2019

v_Enhance

Apr 5, 2023

J

H

k i Zero tolerance

ZetaX 49

N May 4, 2019 by NoDealsHere

Source: Use your common sense! (enough is enough)

Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:

To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.

More specifically:

For new threads:

a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"

b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".

c) Good problem statement:
Some recent really bad post was:
[quote] $lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$ [/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.

For answers to already existing threads:

d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$ , do not answer with " $x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like " $x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.

To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!

Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.

49 replies

ZetaX

Feb 27, 2007

NoDealsHere

May 4, 2019

J

H

Erasing the difference of two numbers

BR1F1SZ 2

N 6 minutes ago by sami1618

Source: Austria National MO Part 1 Problem 3

Consider the following game for a positive integer $n$ . Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$ , or $6$ as $17 - 11=6$ , or $11$ as $17 - 6=11$ .)

For which values of $n$ is it possible to end with only one number remaining on the board?

(Michael Reitmeir)

2 replies

BR1F1SZ

Yesterday at 9:48 PM

sami1618

6 minutes ago

J

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GCD of terms in a sequence

BBNoDollar 0

an hour ago

Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .

0 replies

BBNoDollar

an hour ago

0 replies

J

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Number Theory

fasttrust_12-mn 13

N an hour ago by KTYC

Source: Pan African Mathematics Olympiad P1

Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number

13 replies

fasttrust_12-mn

Aug 15, 2024

KTYC

an hour ago

J

H

GCD of terms in a sequence

BBNoDollar 0

an hour ago

Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .

0 replies

BBNoDollar

an hour ago

0 replies

J

H

Aime type Geo

ehuseyinyigit 3

N an hour ago by sami1618

Source: Turkish First Round 2024

In a scalene triangle $ABC$ , let $M$ be the midpoint of side $BC$ . Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$ . If $(BMN)$ is tangent to $AB$ at $B$ , find $AB/MA$ .

3 replies

ehuseyinyigit

Yesterday at 9:04 PM

sami1618

an hour ago

J

H

minimizing sum

gggzul 1

N 3 hours ago by RedFireTruck

Let $x, y, z$ be real numbers such that $x^2+y^2+z^2=1$ . Find
$min\{12x-4y-3z\}.$

1 reply

gggzul

4 hours ago

RedFireTruck

3 hours ago

J

H

Equilateral Triangle inside Equilateral Triangles.

abhisruta03 2

N 3 hours ago by Reacheddreams

Source: ISI 2021 P6

If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$ , show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$ .

2 replies

abhisruta03

Jul 18, 2021

Reacheddreams

3 hours ago

J

H

USAMO 1984 Problem 5 - Polynomial of degree 3n

Binomial-theorem 8

N 3 hours ago by Assassino9931

Source: USAMO 1984 Problem 5

$P(x)$ is a polynomial of degree $3n$ such that

$\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}$

Determine $n$ .

8 replies

Binomial-theorem

Aug 16, 2011

Assassino9931

3 hours ago

J

H

Finding positive integers with good divisors

nAalniaOMliO 2

N 3 hours ago by KTYC

Source: Belarusian National Olympiad 2025

For every positive integer $n$ write all its divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$ .
Find all $n$ such that $2025 \cdot n=d_{20} \cdot d_{25}$ .

2 replies

nAalniaOMliO

Mar 28, 2025

KTYC

3 hours ago

J

H

Balkan MO 2025 p1

Mamadi 1

N 3 hours ago by KevinYang2.71

Source: Balkan MO 2025

An integer $n > 1$ is called good if there exists a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$ , such that:

$a_i$ and $a_{i+1}$ have different parities for every $1 \le i \le n - 1$

the sum $a_1 + a_2 + \dots + a_k$ is a quadratic residue modulo $n$ for every $1 \le k \le n$

Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

Remark: Here an integer $z$ is considered a quadratic residue modulo $n$ if there exists an integer $y$ such that $y^2 \equiv z \pmod{n}$ .

1 reply

Mamadi

5 hours ago

KevinYang2.71

3 hours ago

J

H

Random Points = Problem

kingu 4

N 3 hours ago by zuat.e

Source: Chinese Geometry Handout

Let $ABC$ be a triangle. Let $\omega$ be a circle passing through $B$ intersecting $AB$ at $D$ and $BC$ at $F$ . Let $G$ be the intersection of $AF$ and $\omega$ . Further, let $M$ and $N$ be the intersections of $FD$ and $DG$ with the tangent to $(ABC)$ at $A$ . Now, let $L$ be the second intersection of $MC$ and $(ABC)$ . Then, prove that $M$ , $L$ , $D$ , $E$ and $N$ are concyclic.

4 replies

kingu

Apr 27, 2024

zuat.e

3 hours ago

J

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CooL geo

Pomegranat 2

N 3 hours ago by Curious_Droid

Source: Idk

In triangle $ABC$ , $D$ is the midpoint of $BC$ . $E$ is an arbitrary point on $AC$ . Let $S$ be the intersection of $AD$ and $BE$ . The line $CS$ intersects with the circumcircle of $ACD$ , for the second time at $K$ . $P$ is the circumcenter of triangle $ABE$ . Prove that $PK \perp CK$ .

2 replies

Pomegranat

Yesterday at 5:57 AM

Curious_Droid

3 hours ago

J

H

Unique Rational Number Representation

abhisruta03 18

N 3 hours ago by Reacheddreams

Source: ISI 2021 P3

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$ .

18 replies

abhisruta03

Jul 18, 2021

Reacheddreams

3 hours ago

J

H

Math solution

Techno0-8 1

N 4 hours ago by jasperE3

Solution

1 reply

Techno0-8

Today at 5:36 PM

jasperE3

4 hours ago

J