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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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
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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
J
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A cyclic inequality
KhuongTrang   3
N 15 minutes ago by BySnipez
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
3 replies
KhuongTrang
Apr 21, 2025
BySnipez
15 minutes ago
J
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Perfect polynomials
Phorphyrion   5
N an hour ago by Davdav1232
Source: 2023 Israel TST Test 5 P3
Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called perfect if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?
5 replies
Phorphyrion
Mar 23, 2023
Davdav1232
an hour ago
J
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Finding all integers with a divisibility condition
Tintarn   14
N an hour ago by Assassino9931
Source: Germany 2020, Problem 4
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
14 replies
Tintarn
Jun 22, 2020
Assassino9931
an hour ago
J
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Geometry Handout is finally done!
SimplisticFormulas   2
N 2 hours ago by parmenides51
If there’s any typo or problem you think will be a nice addition, do send here!
handout, geometry
2 replies
SimplisticFormulas
Today at 4:58 PM
parmenides51
2 hours ago
J
H
Solving the equation n(a^2)=b^2 ; n is not a perfect square
adityaguharoy   0
Apr 12, 2017
If $n$ is a positive integer that is not a perfect square then show that $$n \cdot a^2=b^2$$has no solution with $(a,b) \in \mathbb{N}^2$.

Proof
0 replies
adityaguharoy
Apr 12, 2017
0 replies
J
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Number of Real solution in x of eqn:a^x+b^x=c^x(0<a<b<c)
adityaguharoy   0
Apr 12, 2017
Given real numbers $0<a<b<a+b<c$.
Solve $$a^x+b^x=c^x$$.

Solution

The more General Case
0 replies
adityaguharoy
Apr 12, 2017
0 replies
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No more topics!
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Inscribed quadrilateral
3333   1
N Aug 3, 2011 by oneplusone
Source: St Petersburg 2008 10th grade #2
Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$.

Fresh translation.
1 reply
3333
Jul 28, 2011
oneplusone
Aug 3, 2011
J
Inscribed quadrilateral
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Reveal topic tags
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geometry
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Source: St Petersburg 2008 10th grade #2
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3333
667 posts
3333
#1 h  Jul 28, 2011, 11:28 PM • 1 Y
Y by Adventure10
Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$.

Fresh translation.
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oneplusone
1459 posts
oneplusone
#2 h  Aug 3, 2011, 10:24 AM • 1 Y
Y by Adventure10
Angle chasing gives $\angle ABC=125$ and $\angle ADC=55$. Let $CB$ meet $DA$ at $E$, then $\angle AEB=55$. So $AB+AD=DE>DC$. Or even better, $AB+AD=\frac{\sin 110}{\sin 55}DC$.
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