plane passed through 3 midpoints of edges in a cube, area wanted

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0 replies

jlacosta

Apr 2, 2025

0 replies

Neuberg Cubic leads to fixed point

YaoAOPS 1

N 11 minutes ago by huoxy1623

Source: own

Let $P$ be a point on the Neuberg cubic. Show that as $P$ varies, the Nine Point Circle of the antipedal triangle of $P$ goes through a fixed point.

1 reply

YaoAOPS

an hour ago

huoxy1623

11 minutes ago

Japan MO finals 2023 NT

EVKV 1

N 12 minutes ago by EVKV

Source: Japan MO finals 2023

Determine all positive integers $n$ such that $n$ divides $\phi(n)^{d(n)}+1$ but $d(n)^5$ does not divide $n^{\phi(n)}-1$ .

1 reply

EVKV

an hour ago

EVKV

12 minutes ago

Prove that 4p-3 is a square - Iran NMO 2005 - Problem1

sororak 21

N 34 minutes ago by Maximilian113

Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$ . Prove that $4p-3$ is a perfect square.

21 replies

sororak

Sep 21, 2010

Maximilian113

34 minutes ago

Existence of m and n

shobber 6

N an hour ago by Rayanelba

Source: Pan African 2004

Do there exist positive integers $m$ and $n$ such that:
$3n^2+3n+7=m^3$

6 replies

shobber

Oct 4, 2005

Rayanelba

an hour ago

No more topics!

plane passed through 3 midpoints of edges in a cube, area wanted

parmenides51 1

N Oct 1, 2018 by ythomashu

Source: V.A. Yasinsky Geometry Olympiad 2017 advanced X-XI p5 [Ukranie]

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$ , when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$ .

1 reply

parmenides51

Sep 23, 2018

ythomashu

Oct 1, 2018

plane passed through 3 midpoints of edges in a cube, area wanted

G H J

Reveal topic tags

3D geometry

3-Dimensional Geometry

cube

area

geometry

G H BBookmark kLocked kLocked NReply

Source: V.A. Yasinsky Geometry Olympiad 2017 advanced X-XI p5 [Ukranie]

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parmenides51

30630 posts

parmenides51

#1 h Sep 23, 2018, 8:33 AM • 2 Y

Y by Adventure10, Mango247

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$ , when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$ .

Z K Y

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ythomashu

6322 posts

ythomashu

#2 h Oct 1, 2018, 8:00 PM • 2 Y

Y by Adventure10, Mango247

sol(unfinished

Let the cross section formed by the plane be $PQRST$ and it looks like this(orthographic projection on plane $ABCD$ ):

$[asy]unitsize(50);dot('$C,S$',(0,0),S); fill((.5,1.5)--(-.5,1.5)--(-1,1)--(0,0)--(1,1)--cycle,gray); dot('$D,T$',(-1,1),W); dot('$A$',(0,2),N); dot('$B,R$',(1,1),E); draw((0,0)--(1,1)--(0,2)--(-1,1)--cycle); dot("$Q$",(.5,1.5),NE); dot("$P$",(-.5,1.5),NW); draw((0,0)--(0,1.5),Arrows); label("$\sqrt{\frac{11}8}$",(0,.75),E); [/asy]$

This post has been edited 2 times. Last edited by ythomashu, Oct 1, 2018, 8:49 PM

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