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k a April Highlights and 2025 AoPS Online Class Information

jlacosta 0

Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies

jlacosta

Apr 2, 2025

0 replies

Inspired by old results

sqing 4

Source: Own

Let $a,b>0.$ Prove that
$\frac{(a+1)^2}{b}+\frac{(b+k)^2}{a} \geq4(k+1)$ Where $k\geq 0.$
$\frac{a^2}{b}+\frac{(b+1)^2}{a} \geq4$

4 replies

1 viewing

sqing

5 hours ago

sqing

8 minutes ago

Help with math problem

Glist 0

1. The infinite Morse sequence of zeros and ones, 011010011001..., is constructed as follows: start with 0, then at each step, append a block of the same length as the current sequence, obtained by replacing 0 with 1 and vice versa in the existing block. Is this sequence periodic?
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".

0 replies

Glist

an hour ago

0 replies

Math problem

Glist 3

Given six distinct points on a plane, all pairwise distances between which are different. Prove that there exists a line segment connecting two of these points which is the longest side in one triangle formed by three of the points, and the shortest side in another triangle formed by three of the points.

3 replies

Glist

Yesterday at 2:19 PM

Glist

an hour ago

interesting combinatorics EGMO P5

aditya21 21

Source: EGMO 2015, Problem 5

Let $m, n$ be positive integers with $m > 1$ . Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers.
Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$ .

21 replies

aditya21

Apr 17, 2015

endless_abyss

an hour ago

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

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College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

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k a April Highlights and 2025 AoPS Online Class Information

jlacosta 0

0 replies

jlacosta

Apr 2, 2025

0 replies

Putnam 2019 A5

hoeij 17

Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$ . Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$ , and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$ . Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$ .

17 replies

hoeij

Dec 10, 2019

Ilikeminecraft

2 hours ago

Projection of angle into planes

geekmath-31 0

Question:
consider the angle formed by 2 half lines in the three dimensional space. Prove that the average of the projection of the angle into all of the planes is equal to the angle

The answer is in the attachments.

Please could anyone prove the answer to me in detail.

0 replies

geekmath-31

2 hours ago

0 replies

Projection of angle into planes

geekmath-31 0

0 replies

geekmath-31

3 hours ago

0 replies

Soviet Union University Mathematical Contest

geekmath-31 0

Given a n*n matrix A, prove that there exists a matrix B such that ABA = A

Solution: I have submitted the attachment

The answer is too symbol dense for me to understand the answer.
What I have undertood:

There is use of direct product in the orthogonal decomposition. The decomposition is made with kernel and some T (which the author didn't mention) but as per orthogonal decomposition it must be its orthogonal complement.

Can anyone explain the answer in much much more detail with less use of symbols ( you can also use symbols but clearly define it).

Also what is phi | T ?

0 replies

geekmath-31

4 hours ago

0 replies

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Let $0\le x_{i,j} \le 1$ , where $i=1,2, \ldots m$ and $j=1,2, \ldots n$ . Prove the inequality

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From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$ .In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$ , which intersects $PA$ and $PB$ in $C$ and $D$ . Prove that $M$ is the midpoint $CD$ .

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Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$ .

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k a April Highlights and 2025 AoPS Online Class Information

jlacosta 0

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jlacosta

Apr 2, 2025

0 replies

Introducing myself at AoPS, and what's your magic wand?

asuth_asuth 918

Hi!

I'm Andrew Sutherland. I'm the new Chief Product Officer at AoPS. As you may have read, Richard is retiring and Ben Kornell and I are working together to lead the company now. I'm leading all the software and digital stuff at AoPS. I just wanted to say hello and introduce myself! I'm really excited to be part of the special community that is AoPS.

Previously, I founded Quizlet as a 15-year-old high school student. I did Course 6 at MIT and then left to lead Quizlet full-time for a total of 14 years. I took a few years off and now I'm doing AoPS! I wrote more about all that on my blog: https://asuth.com/im-joining-aops

I have a question for all of you. If you could wave a magic wand, and change anything about AoPS, what would it be? All suggestions welcome! Thank you.

918 replies

asuth_asuth

Mar 30, 2025

Kawhi2

5 hours ago

LaTeX TeXeR PDF exporting error

Bigtree 2

when I try to export the following document as a PDF it says “ Error compiling LaTeX. ! LaTeX Error: Unicode character   (U+202F) not set up for use with LaTeX. ... l.96 ...the even integers from $-180 \text{ to}  360$ inclusive: ! ==> Fatal error occurred, no output PDF file produced!” here’s the document https://artofproblemsolving.com/texer/ngfubjhq

2 replies

Bigtree

Yesterday at 10:47 PM

Bigtree

5 hours ago

k Question

jjmmxx 2

"New users are not allowed to post images in the Community."
Can you tell me about this...?

2 replies

jjmmxx

Yesterday at 11:13 AM

jjmmxx

Yesterday at 11:19 AM

k bar color

jusluo 6

im wondering why my progress bar hasnt turned to a different shade of color after hitting a milestone. I passed orange, and it didn't turn to that color, its still grey (this is in week 12.) is there something I can do to fix it?

6 replies

jusluo

Thursday at 7:19 PM

jkim0656

Thursday at 10:13 PM

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Vietnam Team Selection Test for 2025 IMO

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