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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 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
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Angles, similar triangles, geometry problem
smalkaram_3549   0
4 minutes ago
Completely stuck on this problem.
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smalkaram_3549
4 minutes ago
0 replies
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Conditional maximum
giangtruong13   2
N 7 minutes ago by sqing
Source: Specialized Math
Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$. Find the maximum: $$A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$$
2 replies
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giangtruong13
Mar 22, 2025
sqing
7 minutes ago
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inequality ( 4 var
SunnyEvan   3
N 19 minutes ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
3 replies
SunnyEvan
Yesterday at 5:19 AM
SunnyEvan
19 minutes ago
J
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Inspired by JK1603JK
sqing   16
N 24 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$
16 replies
sqing
Yesterday at 3:31 AM
sqing
24 minutes ago
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Kazakhstan NMO 2009 11 grade P 2
Ovchinnikov Denis   1
N Sep 5, 2010 by Luis González
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$.
Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.
1 reply
Ovchinnikov Denis
Sep 5, 2010
Luis González
Sep 5, 2010
J
Kazakhstan NMO 2009 11 grade P 2
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Reveal topic tags
geometry
perpendicular bisector
geometry proposed
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Ovchinnikov Denis
470 posts
Ovchinnikov Denis
#1 h  Sep 5, 2010, 5:32 PM • 2 Y
Y by Adventure10, Mango247
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$.
Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.
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Luis González
4145 posts
Luis González
#2 h  Sep 5, 2010, 7:52 PM • 1 Y
Y by Adventure10
The key is to show that $B_2$ is equidistant from $I_1,I_2.$ For instance, a step of this proof (see post #2) shows that $B_2$ lies on the perpendicular bisector of segment $I_1I_2$ $\Longrightarrow$ $B_2I_1=B_2I_2.$ Since line $AC \equiv B_1B_2$ is the external bisector of $\angle I_1B_1I_2,$ it follows that $B_2$ lies on $\odot(I_1I_2B_1)$ and it's the midpoint of the arc $I_1B_1I_2.$
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