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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 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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Geo Mock #10
Bluesoul   2
N 6 minutes ago by Bluesoul
Consider acute $\triangle{ABC}$ with $AB=10$, $AC<BC$ and area $135$. The circle $\omega$ with diameter $AB$ meets $BC$ at $E$. Let the orthocenter of the triangle be $H$, connect $CH$ and extend to meet $\omega$ at $N$ such that $NC>HC$ and $NE$ is the diameter of $\omega$. Draw the circumcircle $\Gamma$ of $\triangle{AHB}$, chord $XY$ of $\Gamma$ is tangent to $\omega$ and it passes through $N$, compute $XY$.
2 replies
Bluesoul
Apr 1, 2025
Bluesoul
6 minutes ago
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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   2
N 2 hours ago by Shiyul
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
2 replies
Shiyul
5 hours ago
Shiyul
2 hours ago
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Inequalities
sqing   4
N 5 hours ago by sqing
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
4 replies
sqing
Apr 1, 2025
sqing
5 hours ago
J
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Geo Mock #9
Bluesoul   1
N 5 hours ago by vanstraelen
Consider $\triangle{ABC}$ with $AB=12, AC=22$. The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$. Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$. Extend $AT$ to $BC$ at $R$, given $AR=16$, find $[ABC]$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
5 hours ago
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AB-DE=EF-BC=CD-FA in equiangular 6gon (2017 Kurchatov Olympiad 7 p5 - Russia)
parmenides51   1
N Mar 8, 2021 by ezpotd
All angles of a convex hexagon $ABCDEF$ are equal. Prove that $AB - DE = EF - BC = CD - FA$.
1 reply
parmenides51
Mar 8, 2021
ezpotd
Mar 8, 2021
J
AB-DE=EF-BC=CD-FA in equiangular 6gon (2017 Kurchatov Olympiad 7 p5 - Russia)
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Reveal topic tags
geometry
hexagon
equal angles
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parmenides51
30629 posts
parmenides51
#1 h  Mar 8, 2021, 3:49 PM
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All angles of a convex hexagon $ABCDEF$ are equal. Prove that $AB - DE = EF - BC = CD - FA$.
This post has been edited 1 time. Last edited by parmenides51, Mar 8, 2021, 3:49 PM
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ezpotd
1251 posts
ezpotd
#2 h  Mar 8, 2021, 4:43 PM
Y by
Extend the hexagon to form an equilateral triangle with side length $s$ such that $AB.CD.EF$ are all part of the base.

We wish to prove $AB + BC = EF + DE$, and its cyclic variants.
Let the meeting of the extensions of $AB,EF$ be $X$. Let the meeting of $CD, EF$ be $Y$. Since $AFX,DEY$ are equilateral, $XF + EF + EY = AF + EF + DE = s$. Similarly, letting the intersection of $CD,AB$ be $Z$, we find that $AX + AB + BZ = AF + AB + BC = s$.

Then since $AF + AB + BC = AF + EF + DE$, we have that $AB + BC = EF + DE$. We can prove its cyclic variants using the same strategy.
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