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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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Function equation
luci1337   3
N 10 minutes ago by jasperE3
find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
3 replies
luci1337
Yesterday at 3:01 PM
jasperE3
10 minutes ago
J
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Circumcenter of reflection of collinear points over sides
a1267ab   27
N 10 minutes ago by Giant_PT
Source: USA TST 2025
Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be collinear points such that $AY=AZ$, $BZ=BX$, and $CX=CY$. Points $X'$, $Y'$, and $Z'$ are the reflections of $X$, $Y$, and $Z$ over $BC$, $CA$, and $AB$, respectively. Prove that if $X'Y'Z'$ is a nondegenerate triangle, then its circumcenter lies on the circumcircle of $ABC$.

Michael Ren
27 replies
a1267ab
Jan 11, 2025
Giant_PT
10 minutes ago
J
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a+b+c=abc
KhuongTrang   1
N 19 minutes ago by KhuongTrang
Source: own
Problem. Let $a,b,c$ be three positive real numbers satisfying $a+b+c=abc.$ Prove that$$\sqrt{a^2+b^2+3}+\sqrt{b^2+c^2+3}+\sqrt{c^2+a^2+3}\ge4\cdot \frac{a^2b^2c^2-3}{ab+bc+ca-3}-7.$$There is a very elegant proof :-D Could anyone think of it?
1 reply
1 viewing
KhuongTrang
Wednesday at 11:51 AM
KhuongTrang
19 minutes ago
J
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multiple of 15-15 positive factors
britishprobe17   0
24 minutes ago
Source: KTOM Maret 2025
Find the sum of all natural numbers $n$ such that $n$ is a multiple of $15$ and has exactly $15$ positive factors.
0 replies
britishprobe17
24 minutes ago
0 replies
J
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constant sum PM+PN inside a convex ABCD with AB=CD, MN<=BD
parmenides51   2
N Mar 20, 2023 by huashiliao2020
Source: Greece JBMO TST 2000 p2
Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that:
a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$.
b) $MN\le BD$. When the equality holds?
2 replies
parmenides51
Jun 17, 2019
huashiliao2020
Mar 20, 2023
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constant sum PM+PN inside a convex ABCD with AB=CD, MN<=BD
G H J
Reveal topic tags
geometry
constant
convex quadrilateral
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G H BBookmark kLocked kLocked NReply
Source: Greece JBMO TST 2000 p2
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parmenides51
30629 posts
parmenides51
#1 h  Jun 17, 2019, 9:37 AM • 1 Y
Y by Adventure10
Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that:
a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$.
b) $MN\le BD$. When the equality holds?
Z K Y
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socrates
2105 posts
socrates
#2 h  May 7, 2020, 11:36 PM
Y by
Bumping this problem!
Z K Y
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huashiliao2020
1292 posts
huashiliao2020
#3 h  Mar 20, 2023, 5:34 PM
Y by
a) By similar triangles on PMD and BAD, PM/AB=DP/BD => PM=AB*DP/BD. Similarly, PN=PB*CD/BD=PB*AB/BD. PM+PN=AB(DP+PB)/BD=AB*BD/BD=AB, which is the constant that we seek.
b) I am not sure how to approach this... I know MN <= (less than or equal to) PM+PN=AB, but it is not necessarily true that AB <= BD, right? Unless there is a way to prove that, or a different way to go about this..
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