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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
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Hard geometry
Lukariman   0
8 minutes ago
Given triangle ABC, a line d intersects the sides AB, AC and the line BC at D, E, F respectively.

(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.

(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.

(c) Let $O_a$,$ O_b$, $O_c$ be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle $O_a$$O_b$$O_c$ belongs to d.

(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
0 replies
Lukariman
8 minutes ago
0 replies
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Interesting inequalities
sqing   2
N 13 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , ab+bc+kca =k+2.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{k^2+k+2+2k\sqrt{k+2}}{(k+1)^2}$$Where $ k\in N^+.$
Let $ a,b,c\geq 0 , ab+bc+ca =3.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq 1+\frac{\sqrt 3}{2}$$Let $ a,b,c\geq 0 , ab+bc+2ca =4.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{16}{9}$$
2 replies
1 viewing
sqing
Yesterday at 1:44 PM
sqing
13 minutes ago
J
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CGMO5: Carlos Shine's Fact 5
v_Enhance   61
N 18 minutes ago by Adywastaken
Source: 2012 China Girl's Mathematical Olympiad
As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.
IMAGE
61 replies
v_Enhance
Aug 13, 2012
Adywastaken
18 minutes ago
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Grid combi with T-tetrominos
Davdav1232   1
N 26 minutes ago by NO_SQUARES
Source: Israel TST 8 2025 p1
Let \( f(N) \) denote the maximum number of \( T \)-tetrominoes that can be placed on an \( N \times N \) board such that each \( T \)-tetromino covers at least one cell that is not covered by any other \( T \)-tetromino.

Find the smallest real number \( c \) such that
\[ f(N) \leq cN^2 \]for all positive integers \( N \).
1 reply
Davdav1232
Thursday at 8:29 PM
NO_SQUARES
26 minutes ago
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Root comparing by Viete
giangtruong13   1
N Apr 14, 2025 by RagvaloD
Given equation $ax^2+bx+c=0$ has 2 roots $m,n$ and equation $cx^2+dx+a=0$ has 2 roots called $p,q$. Prove that $$m^2+n^2+p^2+q^2 \geq 4$$
1 reply
giangtruong13
Apr 14, 2025
RagvaloD
Apr 14, 2025
J
Root comparing by Viete
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inequalities
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giangtruong13
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giangtruong13
#1 h  Apr 14, 2025, 3:00 PM
Y by
Given equation $ax^2+bx+c=0$ has 2 roots $m,n$ and equation $cx^2+dx+a=0$ has 2 roots called $p,q$. Prove that $$m^2+n^2+p^2+q^2 \geq 4$$
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RagvaloD
4913 posts
RagvaloD
#2 h  Apr 14, 2025, 5:30 PM
Y by
$m^2+n^2+p^2+q^2 \geq 2|mn|+2|pq| = 2( |\frac{c}{a}|+|\frac{a}{c}|) \geq 4$
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