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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Six variables inequality
JK1603JK   0
an hour ago
Source: unknown
Let $a,b,c,x,y,z>0$ then prove $$\frac{y+z}{x}\cdot \frac{a}{b+c}+\frac{z+x}{y}\cdot \frac{b}{c+a}+\frac{x+y}{z}\cdot \frac{c}{a+b}\ge 2\sqrt{1+\frac{10abc}{(a+b)(b+c)(c+a)}}.$$
0 replies
1 viewing
JK1603JK
an hour ago
0 replies
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2+ab+b^2 =3 $ . Prove that
$$ \frac{53-\sqrt 3}{46} \geq\frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 } \geq \frac{1}{2 }$$$$ \frac{3}{5 }> \frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 } \geq \frac{1}{2 }$$$$ \frac{3}{2} \geq\frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 } +ab\geq \frac{53-\sqrt 3}{46} $$$$ \frac{3}{2} \geq\frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 } +ab\geq \frac{7\sqrt 3-5}{14} $$$$ \frac{53-\sqrt 3}{46} \geq \frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 }-ab\geq -\frac{1}{2 }$$$$ \frac{7\sqrt 3-5}{14}\geq \frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 }-ab\geq -\frac{1}{2 }$$
1 reply
sqing
an hour ago
sqing
an hour ago
J
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Interesting geometry
AlexCenteno2007   1
N 2 hours ago by MathLuis
Let the circle (I) and G be the inscribed circle (with center I) and the centroid of triangle ABC, respectively.
X and X' are the points of tangency of the inscribed circle A-mixtiline and the eccentric circle A-mixtiline with side BC of triangle ABC, respectively.
M is the midpoint of side BC.
K is the intersection of the incircle of circle (I) with BC.
AN is the height of triangle ABC from vertex A.
K' is the reflection of K with respect to M.
L is the second point of intersection between circle ABC (circle of the triangle) and the reflection of line AM with respect to line AI.

NG, XK, LM, and X'K' meet on the circle of ABC.
1 reply
AlexCenteno2007
2 hours ago
MathLuis
2 hours ago
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IMO Shortlist 2014 C4
hajimbrak   25
N 2 hours ago by math-olympiad-clown
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.

Proposed by Tamas Fleiner and Peter Pal Pach, Hungary
25 replies
hajimbrak
Jul 11, 2015
math-olympiad-clown
2 hours ago
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No more topics!
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circumcircle of PBQ passes through circumcenter of ABC, 2MN = AC
parmenides51   1
N Jul 13, 2020 by parmenides51
Source: 2010 Ukraine MO grade IX P3
Given an acute triangle $ABC$. On the perpendicular bisectors of its sides $AB$ and $BC$ respectively, the points $P$ and $Q$ were noted, and $M$ and $N$ were their projections on the side $AC$ (see fig.). It turned out that $2MN = AC$. Prove that the circumcircle of the triangle $PBQ$ passes through the center of the circumcircle of the triangle $ABC$.
IMAGE

(Nagel Igor)
1 reply
parmenides51
Jul 13, 2020
parmenides51
Jul 13, 2020
J
circumcircle of PBQ passes through circumcenter of ABC, 2MN = AC
G H J
Reveal topic tags
geometry
circumcircle
Circumcenter
perpendicular bisector
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G H BBookmark kLocked kLocked NReply
Source: 2010 Ukraine MO grade IX P3
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parmenides51
30629 posts
parmenides51
#1 h  Jul 13, 2020, 6:46 AM • 1 Y
Y by Mango247
Given an acute triangle $ABC$. On the perpendicular bisectors of its sides $AB$ and $BC$ respectively, the points $P$ and $Q$ were noted, and $M$ and $N$ were their projections on the side $AC$ (see fig.). It turned out that $2MN = AC$. Prove that the circumcircle of the triangle $PBQ$ passes through the center of the circumcircle of the triangle $ABC$.
https://cdn.artofproblemsolving.com/attachments/7/e/a1a32217702f761c611e9f8bcf035cf6ea56a5.png

(Nagel Igor)
This post has been edited 2 times. Last edited by parmenides51, Jul 14, 2020, 9:52 PM
Z K Y
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parmenides51
30629 posts
parmenides51
#2 h  Jul 13, 2020, 6:47 AM • 1 Y
Y by Mango247
posted for the image link
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