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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
find question
mathematical-forest   3
N 6 minutes ago by whwlqkd
Are there any contest questions that seem simple but are actually difficult? :-D
3 replies
mathematical-forest
2 hours ago
whwlqkd
6 minutes ago
Midpoints of arcs form a similar triangle
v_Enhance   18
N 20 minutes ago by Stead
Source: USA TSTST 2013, Problem 1
Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
18 replies
v_Enhance
Aug 13, 2013
Stead
20 minutes ago
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   14
N 23 minutes ago by math90
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
14 replies
OgnjenTesic
May 22, 2025
math90
23 minutes ago
MATHirang MATHibay 2025 (Orals, 1.3.2)
BinariouslyRandom   0
35 minutes ago
Find the third smallest number \( n \) which satisfies all the following properties:
\(\begin{array}{ll}
\bullet & n \text{ is a prime number}, \\
\bullet & n \text{ has more than 3 digits when written in base } 3, \\
\bullet & n \text{ has more than one unique digit in base } 3, \\
\bullet & n \text{ is a palindrome in base } 3.
\end{array}\)
Give your answer in base 10.
0 replies
1 viewing
BinariouslyRandom
35 minutes ago
0 replies
Three Distinct Divisors Sum to 2022
ike.chen   31
N 40 minutes ago by Jupiterballs
Source: ISL 2022/N1
A number is called Norwegian if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
31 replies
ike.chen
Jul 9, 2023
Jupiterballs
40 minutes ago
Word problem of Absolute-Value Function
Sukardy   1
N 42 minutes ago by Mathzeus1024
Please help me solve the following word problems. It is about absolute-value function. Any sketch of function's graph will be greatly appreciated.

Problem:
A combat plane is flying $50$ m above the sea. The plane will aim an enemy submarine in the distance of $100$ m. To be aimed on the target, a missile shot by the plane plunges into the sea with a depth of $30$ m. Suppose the surface of the sea to be $X$-axis and the plane's motion is represented by a function $f(x) = p|x|+q$ for $p, q \in \mathbb{R}$. Find the value of $p + q$.
1 reply
Sukardy
Nov 6, 2019
Mathzeus1024
42 minutes ago
Functional Equations
burntpizza001   1
N 2 hours ago by alexheinis
Let $f:R\rightarrow{}R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)$, $\forall x\in R$.

(i) If $f(0)  = 5$ then minimum possible number of values of $x$ satisfying $f(x)=5$, for $x\in[0,170]$ is?

(ii) Graph of $y=f(x)$$ is symmetrical about which line?
1 reply
burntpizza001
May 25, 2025
alexheinis
2 hours ago
Binomial Sum
P162008   1
N 2 hours ago by alexheinis
If $f(x,n) = (x + n)^n - \binom{n}{1} (x + n - 1)^n + \binom{n}{2} (x + n - 2)^n - \cdots (-1)^n x^n$ then

A) $\sum_{r=1}^{4} f(4,r) = 33$

B) $\sum_{r=1}^{4} f(r,4) = 96$

C) $\sum_{r=1}^{5} f(5,r) = 129$

D) $\sum_{r=1}^{5} f(r,5) = 258$
1 reply
P162008
May 26, 2025
alexheinis
2 hours ago
AP calc?
Thayaden   31
N 3 hours ago by Mathzeus1024
How are we all feeling on AP calc guys?
31 replies
Thayaden
May 20, 2025
Mathzeus1024
3 hours ago
Inequalities
sqing   3
N 4 hours ago by sqing
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$
3 replies
sqing
Yesterday at 2:47 AM
sqing
4 hours ago
How can i prove this?
marxs01   1
N 5 hours ago by alexheinis
If $\frac{\cos{x}}{\cos{y}} + \frac{\sin{x}}{\sin{y}} = -1$ prove that $\frac{\cos^3{y}}{\cos{x}} + \frac{\sin^3{y}}{\sin{x}} = 1$
1 reply
marxs01
Yesterday at 10:56 PM
alexheinis
5 hours ago
Inequalities
sqing   1
N Today at 3:01 AM by sqing
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq  43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq  65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq  91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq  122$$
1 reply
sqing
Yesterday at 2:31 PM
sqing
Today at 3:01 AM
Can anyone help me with this inequality problem
a9opsow_   3
N Today at 1:38 AM by sqing
Let a, b, c be nonnegative real numbers, not all equal. Prove that

\frac{(a - bc)^2 + (b - ca)^2 + (c - ab)^2}{(a - b)^2 + (b - c)^2 + (c - a)^2} \geq \frac{1}{2}.
3 replies
a9opsow_
Yesterday at 11:50 AM
sqing
Today at 1:38 AM
Inequalities
sqing   21
N Today at 12:37 AM by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc +  abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc  + \frac{2}{5}abc  \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc  + \frac{7}{20}abc  \leq \frac{2381411}{26460}$$
21 replies
sqing
May 21, 2025
sqing
Today at 12:37 AM
point construction, intersections of cevians with circumcircle, equilateral
parmenides51   1
N May 12, 2023 by Bexultan
Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p5
The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle.

(A. Zaslavsky)
1 reply
parmenides51
Oct 16, 2020
Bexultan
May 12, 2023
point construction, intersections of cevians with circumcircle, equilateral
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Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p5
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parmenides51
30653 posts
#1 • 1 Y
Y by Mango247
The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle.

(A. Zaslavsky)
Z K Y
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Bexultan
178 posts
#2
Y by
Construct a circle passing through $A$ and $C$ such that for any point $P$ of the circle which lies on the same side of the plane as $B$ with respect to $AC$ $\angle APC=60^{\circ}+\beta$. Consider an analogous circle passing through $A$ and $B$. Let $P$ be the point of their intersection. Then $\angle APC=60^{\circ}+\beta$; $\angle BPC=60^{\circ}+\gamma$; $\angle APB=60^{\circ}+\gamma$. It's not hard to see that this point suffices the conditions
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