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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)!

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[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
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Orthocenter lies on circumcircle
whatshisbucket   89
N 26 minutes ago by Mathandski
Source: 2017 ELMO #2
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$

Proposed by Michael Ren
89 replies
whatshisbucket
Jun 26, 2017
Mathandski
26 minutes ago
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Hard math inequality
noneofyou34   5
N 34 minutes ago by JARP091
If a,b,c are positive real numbers, such that a+b+c=1. Prove that:
(b+c)(a+c)/(a+b)+ (b+a)(a+c)/(c+b)+(b+c)(a+b)/(a+c)>= Sqrt.(6(a(a+c)+b(a+b)+c(b+c)) +3
5 replies
noneofyou34
Sunday at 2:00 PM
JARP091
34 minutes ago
J
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Interesting inequalities
sqing   0
37 minutes ago
Source: Own
Let $ a,b>0 $. Prove that
$$\frac{ab-1} {ab(a+b+2)} \leq \frac{1} {8}$$$$\frac{2ab-1} {ab(a+b+1)} \leq 6\sqrt 3-10$$
0 replies
sqing
37 minutes ago
0 replies
J
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Inspired by SXJX (12)2022 Q1167
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 $. Prove that$$\frac{kabc-1} {abc(a+b+c+8(2k-1))}\leq \frac{1}{16 }$$Where $ k>\frac{1}{2}.$
1 reply
sqing
Yesterday at 4:01 AM
sqing
an hour ago
J
No more topics!
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Two circles and Three line concurrency
mofidy   0
Apr 10, 2025
Two circles $W_1$ and $W_2$ with equal radii intersect at P and Q. Points B and C are located on the circles$W_1$ and $W_2$ so that they are inside the circles $W_2$ and $W_1$, respectively. Also, points X and Y distinct from P are located on $W_1$ and $W_2$, respectively, so that:
$$\angle{CPQ} = \angle{CXQ} \text{ and } \angle{BPQ} = \angle{BYQ}.$$The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
0 replies
mofidy
Apr 10, 2025
0 replies
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Two circles and Three line concurrency
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Reveal topic tags
geometry
concurrency
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mofidy
132 posts
mofidy
#1 h  Apr 10, 2025, 3:07 PM
Y by
Two circles $W_1$ and $W_2$ with equal radii intersect at P and Q. Points B and C are located on the circles$W_1$ and $W_2$ so that they are inside the circles $W_2$ and $W_1$, respectively. Also, points X and Y distinct from P are located on $W_1$ and $W_2$, respectively, so that:
$$\angle{CPQ} = \angle{CXQ} \text{ and } \angle{BPQ} = \angle{BYQ}.$$The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
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