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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
1 viewing
jlacosta
May 1, 2025
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\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6
parmenides51   19
N an hour ago by NicoN9
Source: JBMO Shortlist 2017 A1
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
19 replies
parmenides51
Jul 25, 2018
NicoN9
an hour ago
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Inspired by Austria 2025
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 ,a,b\neq 1$ and $ a^2+b^2=1. $ Prove that$$ (a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$$
1 reply
1 viewing
sqing
an hour ago
sqing
an hour ago
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IMO Genre Predictions
ohiorizzler1434   51
N an hour ago by ethan2011
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
51 replies
ohiorizzler1434
May 3, 2025
ethan2011
an hour ago
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Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   17
N 3 hours ago by Ilikeminecraft
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
17 replies
MarkBcc168
Apr 28, 2020
Ilikeminecraft
3 hours ago
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Inequality on area of a triangnle and a quadrilateral (14)
Amir Hossein   1
N Oct 29, 2010 by Martin N.
We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$

Remark. $S$ denotes the area function.
1 reply
Amir Hossein
Oct 29, 2010
Martin N.
Oct 29, 2010
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Inequality on area of a triangnle and a quadrilateral (14)
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Amir Hossein
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Amir Hossein
#1 h  Oct 29, 2010, 10:45 AM • 2 Y
Y by Adventure10, Mango247
We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$

Remark. $S$ denotes the area function.
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Martin N.
434 posts
Martin N.
#2 h  Oct 29, 2010, 4:41 PM • 1 Y
Y by Adventure10
Let $A_1$ and $D_1$ be points on the lines $AB$ and $CD$, respectively, such that $M\in\ A_1D_1$ and $A_1D_1\| BC$. Assume without loss of generality that $A$ lies on the segment $A_1B$. Let $l$ be the line parallel to $BM$ and $CD$ through $A_1$ and denote $X=l\cap AB$. Then, $S_{MD_1B}=S_{MA_1X}\geqslant S_{MA_1A}$. Moreover, because of the parallel lines, we have $S_{BCM}=S_{CD_1M}=S_{A_1BM}$. Consequently, $\left.S_{ABCD}-3S_{BCM}=S_{ABM}+S_{BCM}+S_{CD_1M}+S_{MD_1B}-S_{A_1BM}-S_{BCM}-S_{MCD_1}=S_{MD_1B}-S_{MA_1A}\geqslant 0\right.$, as desired. Equality holds if and only if $AD$ is identical to $A_1D_1$, i.e. iff $AD\|BC$.
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