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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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Romanian National Olympiad 1997 - Grade 9 - Problem 4
Filipjack   1
N 3 minutes ago by navier3072
Source: Romanian National Olympiad 1997 - Grade 9 - Problem 4
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$$$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.
1 reply
Filipjack
Apr 6, 2025
navier3072
3 minutes ago
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I need help with this problem
VIATON   0
20 minutes ago
Let $x,y$ satisfy:
$\frac{4}{(x-1)^2 + (y-2)^2 +4} + \frac{9}{(x-2)^2 + (y-4)^2 +9} = 1$
$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$
Find Max of :$x+y$
0 replies
VIATON
20 minutes ago
0 replies
J
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BMO Shortlist 2021 A5
Lukaluce   17
N 34 minutes ago by jasperE3
Source: BMO Shortlist 2021
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$holds for all $x, y \in \mathbb{R}^{+}$.

Proposed by Nikola Velov, North Macedonia
17 replies
Lukaluce
May 8, 2022
jasperE3
34 minutes ago
J
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Function equation
luci1337   3
N an hour ago by jasperE3
find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
3 replies
luci1337
Yesterday at 3:01 PM
jasperE3
an hour ago
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No more topics!
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3 equal spheres into a cone, ratio wanted
parmenides51   0
Jan 24, 2021
Source: 2015 SPbU finals, grades 10-11 p6 v3 - Saint Petersburg State University School Olympiad
In a cone, whose base diameter is equal to the generatrix, are placed three identical balls, which touch each other externally. Two balls touch the side surface and the base of the cone. The third ball touches the side surface of the cone at a point lying in the same plane with the centers of the balls. Find the ratio of the radii of the base of the cone and the balls.

result
0 replies
parmenides51
Jan 24, 2021
0 replies
J
3 equal spheres into a cone, ratio wanted
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Reveal topic tags
ratio
sphere
cone
3D geometry
geometry
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Source: 2015 SPbU finals, grades 10-11 p6 v3 - Saint Petersburg State University School Olympiad
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parmenides51
30629 posts
parmenides51
#1 h  Jan 24, 2021, 10:09 PM
Y by
In a cone, whose base diameter is equal to the generatrix, are placed three identical balls, which touch each other externally. Two balls touch the side surface and the base of the cone. The third ball touches the side surface of the cone at a point lying in the same plane with the centers of the balls. Find the ratio of the radii of the base of the cone and the balls.

result
This post has been edited 1 time. Last edited by parmenides51, Jan 24, 2021, 10:09 PM
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