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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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S(ai)=S(aj)=S(sigma ai) = n
ilovemath0402   0
a minute ago
Source: Inspired by Romania 1999
Given positive integer $m$. Find all $n$ such that there exist non-negative integer $a_1,a_2,\ldots a_m$ satisfied
$$S(a_1)=S(a_2)=\ldots = S(a_m)=S(a_1+a_2+\ldots + a_m) = n$$P/s: original problem
0 replies
ilovemath0402
a minute ago
0 replies
J
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S(an) greater than S(n)
ilovemath0402   0
5 minutes ago
Source: Inspired by an old result
Find all positive integer $n$ such that $S(an)\ge S(n) \quad \forall a \in \mathbb{Z}^{+}$ ($S(n)$ is sum of digit of $n$ in base 10)
P/s: Original problem
0 replies
ilovemath0402
5 minutes ago
0 replies
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Parallel lines on a rhombus
buratinogigle   0
10 minutes ago
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Given the rhombus $ABCD$ with its incircle $\omega$. Let $E$ and $F$ be the points of tangency of $\omega$ with $AB$ and $AC$ respectively. On the edges $CB$ and $CD$, take points $G$ and $H$ such that $GH$ is tangent to $\omega$ at $P$. Suppose $Q$ is the intersection point of the lines $EG$ and $FH$. Prove that two lines $AP$ and $CQ$ are parallel or coincide.
0 replies
buratinogigle
10 minutes ago
0 replies
J
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Line bisects a segment
buratinogigle   0
19 minutes ago
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Let $ABC$ be a triangle with $AB = AC$. A circle $(O)$ is tangent to sides $AC$ and $AB$, and $O$ is the midpoint of $BC$. Points $E$ and $F$ lie on sides $AC$ and $AB$, respectively, such that segment $EF$ is tangent to circle $(O)$ at point $P$. Let $H$ and $K$ be the orthocenters of triangles $OBF$ and $OCE$, respectively. Prove that line $OP$ bisects segment $HK$.
0 replies
buratinogigle
19 minutes ago
0 replies
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One of those hand-waving proofs
math_explorer   0
Sep 18, 2010
[quote]46337
$A, B, C$ are points on a line in that order. A line through $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, the circle with diameter $AB$ at $R$, and the circle with diameter $BC$ at $S$. Prove that $PR$ = $QS$. (Note that it doesn't matter if $P$ and $Q$ are swapped.)[/quote]

Click to reveal hidden text

The next problem was pseudopseudopseudorandomly selected from Geometry Unsolved Problems, because I skipped the first two numbers, which were both larger than 5000. Ha.
213436
I don't want to type the whole thing, so a link will have to do. My goodness, more terminology.
0 replies
math_explorer
Sep 18, 2010
0 replies
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First spam
math_explorer   2
N Sep 5, 2010 by phiReKaLk6781
I used barycentric coordinates to solve half of a problem. I derived the incenter coordinates and made a bunch of lines and it worked. I read the solution for the other half, not really because I gave up, but because the problem and the solution were next to each other and it was too easy to read.

TRIANGLE CENTERS
Incenter, centroid, circumcenter, orthocenter, nine-point center, Gergonne point, symmedian point, Fermat point. Rooting through Wikipedia references for the Gergonne point you find a paper with 43 theorems, like "The Gergonne point is the perspector of the intouch triangle and the triangle of the reflections of the internal center of similitude of the incenter and the circumcenter in the sides of the excentral triangle." And the scariest bit is I now understand every word there.

THEOREMS I KEEP FORGETTING TO USE
Ptolemy's theorem

Actually, that's it.

STUFF I NEVER GET TO USE
Desargues' theorem
Steiner-Lehmus
Complex number bash
Arrow's theorem (*)

* Yes, it has nothing to do with geometry, or really with anything in the range of high-school contests, but it's awesome.

SOMETHING ELSE
I received my abstract algebra textbook the class I'm in will use. Nothing related to AoPS there, unfortunately. Anyway, much friendlier than last year's.

My math teacher also reports that some of my classmates want to take the AMC 8.

I really wonder too often how much of my personal information a stalker could deduce from this site.

If you read all the way here, please comment just for the heck of it.
2 replies
math_explorer
Sep 4, 2010
phiReKaLk6781
Sep 5, 2010
J
No more topics!
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where a, b, c are positive real numbers
eyesofgod1930   2
N Apr 10, 2025 by sqing
where $a, b, c$ are positive real numbers.Prove that
$\frac{4}{\sqrt{a^{2}+b^{2}+c^{2}+4}}-\frac{9}{\sqrt{(a+b)\sqrt{(a+2c)(b+2c)}}}\leq \frac{5}{8}$
2 replies
eyesofgod1930
Jun 8, 2020
sqing
Apr 10, 2025
J
where a, b, c are positive real numbers
G H J
Reveal topic tags
inequalities
inequalities proposed
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eyesofgod1930
58 posts
eyesofgod1930
#1 h  Jun 8, 2020, 2:06 PM
Y by
where $a, b, c$ are positive real numbers.Prove that
$\frac{4}{\sqrt{a^{2}+b^{2}+c^{2}+4}}-\frac{9}{\sqrt{(a+b)\sqrt{(a+2c)(b+2c)}}}\leq \frac{5}{8}$
Z K Y
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duonghy93
40 posts
duonghy93
#2 h  Jun 8, 2020, 2:37 PM • 1 Y
Y by eyesofgod1930
I set t=a+b+c.Use the derivative f (t) => P max=5/8
Attachments:
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sqing
42564 posts
sqing
#3 h  Apr 10, 2025, 2:51 AM
Y by
Let $a,b,c>0$. Prove that$$\frac{8}{\sqrt{a^2+b^2+c^2+1}} - \frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}} \leq \frac{5}{2}$$Let $a,b,c>0$. Prove that$$\frac{2}{(a+1)(b+1)(c+1)}-\frac{1}{\sqrt{a^2+b^2+c^2+1}}\leq 1$$
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