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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.

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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
0 replies
J
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Angles, similar triangles, geometry problem
smalkaram_3549   0
4 minutes ago
Completely stuck on this problem.
0 replies
smalkaram_3549
4 minutes ago
0 replies
J
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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
1 viewing
giangtruong13
Mar 22, 2025
sqing
7 minutes ago
J
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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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Mixtilinear circle
randomusername   1
N Jul 14, 2014 by IDMasterz
Source: Kürschák 2004, problem 1
Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.
1 reply
randomusername
Jul 13, 2014
IDMasterz
Jul 14, 2014
J
Mixtilinear circle
G H J
Reveal topic tags
geometry
circumcircle
geometry unsolved
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G H BBookmark kLocked kLocked NReply
Source: Kürschák 2004, problem 1
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randomusername
1059 posts
randomusername
#1 h  Jul 13, 2014, 9:51 PM • 2 Y
Y by Adventure10, Mango247
Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.
Z K Y
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IDMasterz
1412 posts
IDMasterz
#2 h  Jul 14, 2014, 5:21 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Its the ex-mixtillinear circle, intraversion is equivalent to the incircle, and this is well known. Either way, $R$ is tangency point of $k$ and $w$, $I_A$ excentre. $RP, RQ$ intersect at midpoints of arcs $BCA, ABC$, let they be $M_C, M_B$. Pascal on $M_BRM_CCAB \implies I_A, P, Q$ are collinear. But, $AI_A \perp PQ$.
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