More questions I have!

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AnikJain3pro5un00bs
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More questions I have!

Post by AnikJain3pro5un00bs »

Hello, friends! :D
Long time no see, ha ha, more than two years since I made post here! You all are very good, and help me out when I need it with the difficult questions!
Since that time, I report happiness with both life and Science Olympiad (you can see this by my picture, it is a smiley face for a reason!), having done a very good job in Scrambler. I am happy with my performance, and am glad to come back after a long time and hope to see many of the same faces that helped me so long ago, when I needed it! :)
After taking a brief stop with the Science Olympiad last year, I come back to see Fermi Questions event happening this year! Ha, very exciting, I do enjoy maths. I hope I will get to do well in the Fermi Question and see many of you again at the national competition!
Hahaha, maybe I chat too long, ok? Now, I say that I have more questions for you: Very hard Math Questions, but I realize hard questions need to be solved in order to make the Math Olympiad Summer Program, which is my goal! So I do hard questions. Maybe too many questions not good, so I give you one at a time, ok? Maybe first problems not so hard, but I appreciate all the help that can be given to me!

Here is the first question that I need help with:
How many ways are there to arrange the numbers 1 through 9 in a 3x3 grid, such that the numbers are increasing from left to right in each row, and increasing from top to bottom in each column?

Again, I am very much appreciative of all of the help you give, and am privileged to call all of you my good friends! :) Looking forward to seeing you at the national olympiad.
:D hoping to enjoy another year of the science olympiad!!
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Re: More questions I have!

Post by alchzh »

Ah! I remember this question. It's a pretty stock question for various math things.

Fast Method (Hook Length)
Basically this question is asking for the number of Standard Young Tableaux with the shape .

Note that there are a total of cells in the tableau (this is ). The hook lengths for this diagram look like this:

Image

We are essentially drawing "hooks" for each cell with the cell being the upper-left member of the hook and going down and right all the way to the ends. Then we count up the cells to get the length. Plugging this into the hook-length formula we get

Slow method

I'm just going to direct you to this solution for when it came up on the 2010 AMC 12B (Credit goes to some AoPS users that aren't me)
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Re: More questions I have!

Post by Schrodingerscat »

alchzh wrote:Ah! I remember this question. It's a pretty stock question for various math things.

Fast Method (Hook Length)
Basically this question is asking for the number of Standard Young Tableaux with the shape .

Note that there are a total of cells in the tableau (this is ). The hook lengths for this diagram look like this:

Image

We are essentially drawing "hooks" for each cell with the cell being the upper-left member of the hook and going down and right all the way to the ends. Then we count up the cells to get the length. Plugging this into the hook-length formula we get

Slow method

I'm just going to direct you to this solution for when it came up on the 2010 AMC 12B (Credit goes to some AoPS users that aren't me)
Reminding me why I quit math again...
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AnikJain3pro5un00bs
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Re: More questions I have!

Post by AnikJain3pro5un00bs »

Hellooooo!!! THank you very much for the help :)
As promised, I have another question!! Is hard, you need to think in MANY dimensions to get the answer!
ok! without further ado,
how many 20-dimensional faces does a 21-dimensional solid have???

Thank you very much!
:D hoping to enjoy another year of the science olympiad!!
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Re: More questions I have!

Post by alchzh »

?

EDIT: I assume you mean cubes right...?
Last edited by alchzh on Fri Jan 05, 2018 4:14 pm, edited 1 time in total.
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Re: More questions I have!

Post by NeilMehta »

AnikJain3pro5un00bs wrote:Hellooooo!!! THank you very much for the help :)
As promised, I have another question!! Is hard, you need to think in MANY dimensions to get the answer!
ok! without further ado,
how many 20-dimensional faces does a 21-dimensional solid have???

Thank you very much!
21 dimensional cube should have 42 20-dimensional faces
EDIT: Whoops, question was answered already, my fault
Last edited by NeilMehta on Fri Jan 05, 2018 4:15 pm, edited 1 time in total.
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Re: More questions I have!

Post by lumosityfan »

All the answers to the questions is 42.
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Re: More questions I have!

Post by AnikJain3pro5un00bs »

Helloooo Jeff, that is a very smart analyze!
however, you make this assumption from two questions! That is not good engineer induction, ha ha ha!
Ok! Another question, maybe answer 42 this time, maybe not...I do not know!

How many different numbers can be represented in both the form and for some ??

Thank you for your time and answers, these questions very hard!
:D hoping to enjoy another year of the science olympiad!!
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Re: More questions I have!

Post by lumosityfan »

42.
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Re: More questions I have!

Post by AnikJain3pro5un00bs »

ah!! but can you prove?
:D hoping to enjoy another year of the science olympiad!!

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