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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
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Geometry books
T.Mousavidin   1
N 12 minutes ago by martianrunner
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,
1 reply
T.Mousavidin
2 hours ago
martianrunner
12 minutes ago
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Inequalities
sqing   16
N an hour ago by martianrunner
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$ \frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$$$$\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
16 replies
sqing
Apr 25, 2025
martianrunner
an hour ago
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Sequence
lgx57   7
N 3 hours ago by jasperE3
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
7 replies
lgx57
Apr 27, 2025
jasperE3
3 hours ago
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Geometry Angle Chasing
Sid-darth-vater   6
N 4 hours ago by sunken rock
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
6 replies
Sid-darth-vater
Apr 21, 2025
sunken rock
4 hours ago
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50 shots on a 70x70 target, 2 have distance <=15 (2005 AMQ Concours Col. p1)
parmenides51   1
N Mar 12, 2021 by kootrapali
During a competition, a shooter received $50$ consecutive shots of a pistol at a square target of $70$ cm by $70$ cm. Prove that at least two of the hole centers on the target are within $15$ cm apart.
1 reply
parmenides51
Mar 12, 2021
kootrapali
Mar 12, 2021
J
50 shots on a 70x70 target, 2 have distance <=15 (2005 AMQ Concours Col. p1)
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Reveal topic tags
geometry
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combinatorial geometry
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parmenides51
30650 posts
parmenides51
#1 h  Mar 12, 2021, 7:04 PM
Y by
During a competition, a shooter received $50$ consecutive shots of a pistol at a square target of $70$ cm by $70$ cm. Prove that at least two of the hole centers on the target are within $15$ cm apart.
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kootrapali
4527 posts
kootrapali
#2 h  Mar 12, 2021, 7:28 PM • 1 Y
Y by judgefan99
Divide the square into $49$ squares with side length $10$. Then there must be a square with $2$ or more points, but these could be at most $10\sqrt{2}<15$ units apart, as desired.
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