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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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jlacosta
Apr 2, 2025
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Inspired by bamboozled
sqing   0
5 minutes ago
Source: Own
Let $ a,b,c $ be reals such that $(a^2+1)(b^2+1)(c^2+1) = 27. $Prove that $$1-3\sqrt 3\leq ab + bc + ca\leq 6$$
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1 viewing
sqing
5 minutes ago
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Range of ab + bc + ca
bamboozled   1
N 14 minutes ago by sqing
Let $(a^2+1)(b^2+1)(c^2+1) = 9$, where $a, b, c \in R$, then the number of integers in the range of $ab + bc + ca$ is __
1 reply
+1 w
bamboozled
28 minutes ago
sqing
14 minutes ago
J
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Functional Equation
AnhQuang_67   4
N 14 minutes ago by AnhQuang_67
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
4 replies
AnhQuang_67
Yesterday at 4:50 PM
AnhQuang_67
14 minutes ago
J
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Inradius and ex-radii
bamboozled   0
22 minutes ago
Let $ABC$ be a triangle and $r, r_1, r_2, r_3$ denote its inradius and ex-radii opposite to the vertices $A, B, C$ respectively. If $a> r_1, b > r_2$ and $c > r_3$, then which of the following is/are true?
(A) $\angle{B}$ is obtuse
(B) $\angle{A}$ is acute
(C) $3r > s$, where $s$ is semi perimeter
(D) $3r < s$, where $s$ is semi perimeter
0 replies
bamboozled
22 minutes ago
0 replies
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AS = BS if <ACB = 45^o, MAKS = 60^o, AK = 2KC, AS _|_ BK
parmenides51   1
N May 31, 2022 by Tsikaloudakis
Source: 2022 SPbU Qualifying round, grades 8-9 p7 - Saint Petersburg State University School Olympiad
On side $AC$ of triangle $ABC$, where $\angle ACB = 45^o$, there is a point $K$ such that $AK = 2KC$. A point $S$ was found on the segment $BK$ such that $AS \perp BK$ and $\angle AKS = 60^o$. Prove that $AS = BS$.
1 reply
parmenides51
May 29, 2022
Tsikaloudakis
May 31, 2022
J
AS = BS if <ACB = 45^o, MAKS = 60^o, AK = 2KC, AS _|_ BK
G H J
Reveal topic tags
geometry
equal segments
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G H BBookmark kLocked kLocked NReply
Source: 2022 SPbU Qualifying round, grades 8-9 p7 - Saint Petersburg State University School Olympiad
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parmenides51
30629 posts
parmenides51
#1 h  May 29, 2022, 11:49 PM
Y by
On side $AC$ of triangle $ABC$, where $\angle ACB = 45^o$, there is a point $K$ such that $AK = 2KC$. A point $S$ was found on the segment $BK$ such that $AS \perp BK$ and $\angle AKS = 60^o$. Prove that $AS = BS$.
Z K Y
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Tsikaloudakis
974 posts
Tsikaloudakis
#2 h  May 31, 2022, 2:51 PM
Y by
SEE THE FIGURE:
Attachments:
This post has been edited 1 time. Last edited by Tsikaloudakis, May 31, 2022, 2:54 PM
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