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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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Geometry
Lukariman   9
N a minute ago by lbh_qys
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
9 replies
+1 w
Lukariman
Tuesday at 12:43 PM
lbh_qys
a minute ago
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I need the technique
DievilOnlyM   15
N an hour ago by sqing
Let a,b,c be real numbers such that: $ab+7bc+ca=188$.
FInd the minimum value of: $5a^2+11b^2+5c^2$
15 replies
DievilOnlyM
May 23, 2019
sqing
an hour ago
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Linear colorings mod 2^n
vincentwant   1
N an hour ago by vincentwant
Let $n$ be a positive integer. The ordered pairs $(x,y)$ where $x,y$ are integers in $[0,2^n)$ are each labeled with a positive integer less than or equal to $2^n$ such that every label is used exactly $2^n$ times and there exist integers $a_1,a_2,\dots,a_{2^n}$ and $b_1,b_2,\dots,b_{2^n}$ such that the following property holds: For any two lattice points $(x_1,y_1)$ and $(x_2,y_2)$ that are both labeled $t$, there exists an integer $k$ such that $x_2-x_1-ka_t$ and $y_2-y_1-kb_t$ are both divisible by $2^n$. How many such labelings exist?
1 reply
vincentwant
Apr 30, 2025
vincentwant
an hour ago
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sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   1
N an hour ago by vincentwant
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases} \sqrt{n}&\text{ if n is a perfect square} \\ n+p&\text{ otherwise} \end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
1 reply
vincentwant
Apr 30, 2025
vincentwant
an hour ago
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No more topics!
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A problem with non-negative a,b,c
KhuongTrang   2
N Mar 29, 2025 by KhuongTrang
Source: own
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
2 replies
KhuongTrang
Mar 4, 2025
KhuongTrang
Mar 29, 2025
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A problem with non-negative a,b,c
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Reveal topic tags
inequalities
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Source: own
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KhuongTrang
731 posts
KhuongTrang
#1 h  Mar 4, 2025, 3:50 PM • 5 Y
Y by NO_SQUARES, jokehim, math90, Zuyong, NguyenVanHoa29
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
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KhuongTrang
731 posts
KhuongTrang
#3 h  Mar 7, 2025, 3:02 PM • 4 Y
Y by jokehim, math90, Zuyong, NguyenVanHoa29
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{8a(b+c)+(b-c)^{2}}+\sqrt{8b(c+a)+(c-a)^{2}}+\sqrt{8c(a+b)+(a-b)^{2}}\ge 12\cdot\frac{ab+bc+ca}{a+b+c}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
MSE
Problem. Let $a,b,c$ be non-negative real variables with $a+b+c\neq 0.$ Prove that $$\color{blue}{\frac{ab(c+a)(c+b)}{2c+a+b}+\frac{bc(a+b)(a+c)}{2a+b+c}+\frac{ca(b+c)(b+a)}{2b+c+a}\ge \frac{(ab+bc+ca)^2}{a+b+c}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ or $(a,b,c)\sim(0,0,t)$ where $t>0.$
This post has been edited 2 times. Last edited by KhuongTrang, Apr 30, 2025, 3:17 PM
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KhuongTrang
731 posts
KhuongTrang
#4 h  Mar 29, 2025, 12:13 AM • 1 Y
Y by NguyenVanHoa29
Problem. Let $a,b,c$ be non-negative real variables. Prove that $$\color{blue}{\sqrt{3(a^2+b^2)+2(ab+bc+ca)}+\sqrt{3(b^2+c^2)+2(ab+bc+ca)}+\sqrt{3(c^2+a^2)+2(ab+bc+ca)}\ge 2\sqrt{3}(a+b+c).}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,0,0)$ where $t\ge 0.$
This post has been edited 2 times. Last edited by KhuongTrang, Mar 29, 2025, 12:15 AM
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