G
Topic
First Poster
Last Poster
Hard 3-d geometry question
ZMB038 0
In tetrahedron 
A cube is inscribed in the tetrahedron so that one of its vertices is at 
and the opposite vertex lies on face 
Let 
and 
Show that the side length of the cube is
![\[\frac{abc}{ab + ac + bc}.\]](//latex.artofproblemsolving.com/9/c/c/9ccb0677c6e6f02e3cabbc48f2f0a6bf32d0d9a2.png)
Can someone please help, I'm not that smart and I just can't solve this :wallbash_red: :wallbash_red:
A cube is inscribed in the tetrahedron so that one of its vertices is at 
and the opposite vertex lies on face 
Let 
and 
Show that the side length of the cube is![\[\frac{abc}{ab + ac + bc}.\]](http://latex.artofproblemsolving.com/9/c/c/9ccb0677c6e6f02e3cabbc48f2f0a6bf32d0d9a2.png)
Can someone please help, I'm not that smart and I just can't solve this :wallbash_red: :wallbash_red:
0 replies
What's the chance that two AoPS accounts generate with the same icon?
Math-lover1 6
N
by ZMB038
So I've been wondering how many possible "icons" can be generated when you first create an account. By "icon" I mean the stack of cubes as the first profile picture before changing it.
I don't know a lot about how AoPS icons generate, so I have a few questions:
- Do the colors on AoPS icons generate through a preset of colors or the RGB (red, green, blue in hexadecimal form) scale? If it generates through the RGB scale, then there may be greater than
different icons.
- Do the arrangements of the stacks of blocks in the icon change with each account? If so, I think we can calculate this through considering each stack of blocks independently.
I don't know a lot about how AoPS icons generate, so I have a few questions:
- Do the colors on AoPS icons generate through a preset of colors or the RGB (red, green, blue in hexadecimal form) scale? If it generates through the RGB scale, then there may be greater than
different icons.- Do the arrangements of the stacks of blocks in the icon change with each account? If so, I think we can calculate this through considering each stack of blocks independently.
6 replies
9 What competitions do you do
VivaanKam 14
N
by ZMB038
I know I missed a lot of other competitions so if you didi one of the just choose "Other".
14 replies
National Team Round Problem 10?!
Leeoz 6
N
by ZMB038
This was the 2015 national team round problem 10...
[quote=The hardest problem in MathCounts]
In the city of Trichotomy, every day the weather is exactly one of the following:
sunny, cloudy or rainy. Each day has a 50% chance of having the same weather
as the day before and a 25% chance of having each of the other two types of
weather. If it does not rain on Friday, what is the probability that there will be
no rain during the weekend (Saturday and Sunday)? Express your answer as a
common fraction.
[/quote]
[quote=The hardest problem in MathCounts]
In the city of Trichotomy, every day the weather is exactly one of the following:
sunny, cloudy or rainy. Each day has a 50% chance of having the same weather
as the day before and a 25% chance of having each of the other two types of
weather. If it does not rain on Friday, what is the probability that there will be
no rain during the weekend (Saturday and Sunday)? Express your answer as a
common fraction.
[/quote]
6 replies
prime numbers
wpdnjs 106
N
by ZMB038
does anyone know how to quickly identify prime numbers?
thanks.
thanks.
106 replies
max number of candies
orangefronted 10
N
by SpeedCuber7
A store sells a strawberry flavoured candy for 1 dollar each. The store offers a promo where every 4 candy wrappers can be exchanged for one candy. If there is no limit to how many times you can exchange candy wrappers for candies, what is the maximum number of candies I can obtain with 100 dollars?
10 replies
9 Have you participated in the MATHCOUNTS competition?
aadimathgenius9 40
N
by sanaops9
Have you participated in the MATHCOUNTS competition before?
40 replies
Easy number theory
britishprobe17 28
N
by ZMB038
The number of factors from 2024 that are greater than 
are
are
28 replies
How to get a 300+ on the NWEA MAP MATH test (URGENT)
nmlikesmath 14
N
by Craftybutterfly
I have 4 days till this test, I'm wondering how do I get a 300+ and what do I need to know, thank you.
14 replies
How many ways to spell the word COUNT
yinglinwu 7
N
by Andrew2019
This is the problem 2.771 from the book Competition Math For Middle School. The printed answer is 16, which is obtained by going through the triangle. I have no idea about what's going on there. Can anyone help explain please?
BTW, my understanding about a way to spell COUNT is simply to choose an order of those letters and write down them one by one at their designated locations, like writing down U first at the 3rd location, then O at the 2nd position, N at 4th, T at 5th, and C at 1st. Therefore, in total we should have 5! ways to spell the word.
BTW, my understanding about a way to spell COUNT is simply to choose an order of those letters and write down them one by one at their designated locations, like writing down U first at the 3rd location, then O at the 2nd position, N at 4th, T at 5th, and C at 1st. Therefore, in total we should have 5! ways to spell the word.
7 replies
