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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
[*]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
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2 var inquality
sqing   5
N 2 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+2ab=4 . $ Prove that
$$ 3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$$$$ 9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$$$$9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$$
5 replies
sqing
Today at 1:50 AM
sqing
2 minutes ago
J
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Extent of Group Theory needed for NT
Math-Problem-Solving   0
11 minutes ago
How much of group theory, knowledge of rings and fields is required for doing number theory in full fledge. Given I am a class 11 high schooler student with a little background in math olympiad, so please mention some resources for learning these things which I can understand.
0 replies
Math-Problem-Solving
11 minutes ago
0 replies
J
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No perfect squares in A-A
JustPostChinaTST   7
N 11 minutes ago by kes0716
Source: 2022 China TST, Test 3 P5
Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.
7 replies
+1 w
JustPostChinaTST
Apr 30, 2022
kes0716
11 minutes ago
J
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Quadric function
soryn   2
N an hour ago by soryn
If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integer solution,prove that M has at most three elements.
2 replies
soryn
Today at 2:47 AM
soryn
an hour ago
J
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AP/AD >= 1 - BC^2/(AB^2 + CA^2) , altitudes related
parmenides51   0
Dec 10, 2021
Source: Indonesia INAMO Shortlist 2009 G6 https://artofproblemsolving.com/community/c1101409_
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$
0 replies
parmenides51
Dec 10, 2021
0 replies
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AP/AD >= 1 - BC^2/(AB^2 + CA^2) , altitudes related
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Reveal topic tags
geometry
geometric inequality
altitudes
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Source: Indonesia INAMO Shortlist 2009 G6 https://artofproblemsolving.com/community/c1101409_
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parmenides51
30629 posts
parmenides51
#1 h  Dec 10, 2021, 6:59 PM
Y by
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$
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