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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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BMO 2024 SL A1
MuradSafarli   8
N 13 minutes ago by ja.
A1.

Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[ \frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z} \]
holds for all positive real numbers \( a, b \) and \( c \).
8 replies
MuradSafarli
Apr 27, 2025
ja.
13 minutes ago
J
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Medium geometry with AH diameter circle
v_Enhance   94
N an hour ago by alexanderchew
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
94 replies
+1 w
v_Enhance
Jun 28, 2016
alexanderchew
an hour ago
J
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problem interesting
Cobedangiu   7
N an hour ago by vincentwant
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
7 replies
Cobedangiu
Yesterday at 5:06 AM
vincentwant
an hour ago
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another problem
kjhgyuio   1
N an hour ago by lpieleanu
........
1 reply
kjhgyuio
2 hours ago
lpieleanu
an hour ago
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No more topics!
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max area of ABCD with sidelengthts 1, 2\sqrt2 , 3 , 4
parmenides51   1
N Mar 18, 2021 by ilovepizza2020
Source: 2020 Moldova NMO 10.8
Find the maximum possible area of the quadrilateral whose sidelengths are equal to $1$ cm, $2\sqrt2$ cm, $3$ cm and $4$ cm.
1 reply
parmenides51
Mar 18, 2021
ilovepizza2020
Mar 18, 2021
J
max area of ABCD with sidelengthts 1, 2\sqrt2 , 3 , 4
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Reveal topic tags
geometry
max
geometric inequality
area of a triangle
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G H BBookmark kLocked kLocked NReply
Source: 2020 Moldova NMO 10.8
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parmenides51
30650 posts
parmenides51
#1 h  Mar 18, 2021, 5:49 PM • 1 Y
Y by samrocksnature
Find the maximum possible area of the quadrilateral whose sidelengths are equal to $1$ cm, $2\sqrt2$ cm, $3$ cm and $4$ cm.
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ilovepizza2020
12156 posts
ilovepizza2020
#2 h  Mar 18, 2021, 5:53 PM • 2 Y
Y by samrocksnature, tenebrine
The maximum area occurs when it is cyclic so it should be $\sqrt{22+12\sqrt{2}} = 2+3\sqrt{2}$
This post has been edited 1 time. Last edited by ilovepizza2020, Mar 18, 2021, 5:53 PM
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