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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)!

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[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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Inspired by Ruji2018252
sqing   1
N a minute ago by lbh_qys
Source: Own
Let $ a,b,c $ be reals such that $ a^2+b^2+c^2-2a-4b-4c=7. $ Prove that
$$ -4\leq 2a+b+2c\leq 20$$$$5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$$$$ 11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$$
1 reply
1 viewing
sqing
4 hours ago
lbh_qys
a minute ago
J
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F=(F^3+F^3)/9-2F^3
Yiyj1   1
N 3 minutes ago by sp0rtman00000
Source: 101 Algebra Problems for the AMSP
Define the Fibonacci sequence $F_n$ as $$F_1=F_2=1, F_{n+1}+F_n=F_{n-1}$$fir $n \in \mathbb{N}$. Prove that $$F_{2n}=\dfrac{F_{2n+2}^3+F_{2n-2}^3}{9}-2F_{2n}^3$$for all $n \ge 2$.
1 reply
1 viewing
Yiyj1
3 hours ago
sp0rtman00000
3 minutes ago
J
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Funky and nice NT FE
yaybanana   0
5 minutes ago
Source: own
find all functions $f: \mathbb{Z}^* \rightarrow \mathbb{Z}^*$ ,s.t :

$f(n)+f(m) \mid n^2+nf(m)$

for all $m,n \in \mathbb{Z}^*$
0 replies
yaybanana
5 minutes ago
0 replies
J
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4 lines concurrent
Zavyk09   3
N 13 minutes ago by ItzsleepyXD
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
3 replies
Zavyk09
Yesterday at 11:51 AM
ItzsleepyXD
13 minutes ago
J
No more topics!
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projections of point P onto lateral faces of pyramid are concyclic
parmenides51   0
Oct 19, 2020
Source: 2007 Oral Moscow Geometry Olympiad grades 10-11 p5
At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle.

(A. Zaslavsky)
0 replies
parmenides51
Oct 19, 2020
0 replies
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projections of point P onto lateral faces of pyramid are concyclic
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Reveal topic tags
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Concyclic
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Source: 2007 Oral Moscow Geometry Olympiad grades 10-11 p5
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parmenides51
30630 posts
parmenides51
#1 h  Oct 19, 2020, 12:20 PM
Y by
At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle.

(A. Zaslavsky)
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