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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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[*]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
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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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Find set of all points for a regular triangular pyramid
orl   4
N Sep 24, 2017 by parmenides51
Source: First Zhautykov Olympiad 2005, Problem 3
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! = S)$ in the space satisfing the equation $ |cos ASD - 2cosBSD - 2 cos CSD| = 3$.
4 replies
orl
Dec 22, 2008
parmenides51
Sep 24, 2017
J
Find set of all points for a regular triangular pyramid
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Reveal topic tags
geometry
3D geometry
pyramid
geometry unsolved
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G H BBookmark kLocked kLocked NReply
Source: First Zhautykov Olympiad 2005, Problem 3
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orl
3647 posts
orl
#1 h  Dec 22, 2008, 12:46 AM • 2 Y
Y by Adventure10, Mango247
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! = S)$ in the space satisfing the equation $ |cos ASD - 2cosBSD - 2 cos CSD| = 3$.
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Vo Duc Dien
341 posts
Vo Duc Dien
#2 h  Mar 14, 2011, 11:49 PM • 2 Y
Y by Adventure10, Mango247
Did you mean (D!= S) for D ≠ S?

Please clarify!
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gold46
595 posts
gold46
#3 h  Jan 14, 2012, 6:36 PM • 2 Y
Y by Adventure10, Mango247
some one could give me solution ?
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ank 1729
272 posts
ank 1729
#4 h  Jul 29, 2017, 3:47 PM • 2 Y
Y by Adventure10, Mango247
Anyone? Very interesting problem in my opinion.
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parmenides51
30629 posts
parmenides51
#5 h  Sep 24, 2017, 9:30 PM • 1 Y
Y by Adventure10
a better restatement of the problem

Let ${SABC}$. be a regular triangular pyramid (${SA=SB=SC}$. and ${AB=BC=CA)}$ ). Find the locus of all points ${D \, (D\ne S)}$. in the space that satisfy the equation ${ |cos \delta_A -2cos \delta_B - 2cos \delta_C | = 3 }$. where the angle ${\delta_X=\angle XSD}$ for each ${X \in \{ A,B,C\} }$.
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