Prelim. marathon? (3 Viewers)

[HSC2014]

Is the answer 3?

Yep I got the same. Can achieve answer using simple algebra manipulation.

Here.
A certain substance doubles its volume every minute. At 9:00am a small amount is placed in a container and at 10:00am the container just fills. What is the time at which the container was one quarter full?

 

[Shazer2]

Spoiler

My solution:


I hope it's right.

 

[Shazer2]

yep i got the same. Can achieve answer using simple algebra manipulation.

Here.
A certain substance doubles its volume every minute. At 9:00am a small amount is placed in a container and at 10:00am the container just fills. What is the time at which the container was one quarter full?

9:15am?

 

[Timske]

Let V = x
At 10am it is full , V = x
At 9:59am it is half full , V = x/2
At 9:58am it is one-quarter full , V = x/4
Therefore at 9:58am the container is one-quarter full

 

[HSC2014]

Let V = x
At 10am it is full , V = x
At 9:59am it is half full , V = x/2
At 9:58am it is one-quarter full , V = x/4
Therefore at 9:58am the container is one-quarter full

Too easy

 

[Wonder]

Here's one:

 

[Shazer2]

Here's one:

View attachment 26514

Indice laws apply here right?

 

[HSC2014]

^ The numerator actually becomes 18^2n

 

[Shazer2]

^ The numerator actually becomes 18^2n

Oh right, so it equals:
?

 

[Indoz]

Oh right, so it equals:
?

yeah, the bases combine but the powers dont, because they're the same

 

[Timske]

Let n be 1
6^2 * 3^2 =/= 18^4
therefore 6^2n * 3^2n =/= 18^4n

 

[Shazer2]

That makes sense! I didn't know powers didn't add when there was a pronumeral.

 

[hamster-lol]

another question to keep marathon rolling (other people need to post questions too lol)

amc 2012

 

[hamster-lol]

a) Let

This is expressed in ascending order. By considering in descending order show that



b) Consider a bag. This bag has every single letter of the alphabet in it, and only one of each letter so that there are 26 letters in the bag. Let the probability of picking the letter 'z' out of the bag be

Now consider another bag, called bag 2. This bag contains 1 A, 2 B's, 3 C's, and so on, up to 26 Z's.

Show that none of the letters in bag 2 have the same probability of being pulled out as

 

[Indoz]

That makes sense! I didn't know powers didn't add when there was a pronumeral.

the powers dont add if:
- the powers are different but the bases are the same (a^n * a^m = a^m+n)
- the bases are the same but the powers are the same/different (m^a * n^a = mn^a)

got nothing to do with the pronumeral

 

[Shazer2]

I get it now.

 

[RealiseNothing]

a) Let

This is expressed in ascending order. By considering in descending order show that



b) Consider a bag. This bag has every single letter of the alphabet in it, and only one of each letter so that there are 26 letters in the bag. Let the probability of picking the letter 'z' out of the bag be

Now consider another bag, called bag 2. This bag contains 1 A, 2 B's, 3 C's, and so on, up to 26 Z's.

Show that none of the letters in bag 2 have the same probability of being pulled out as

Very nice question.

 

[Sy123]

HINT for a)

Express all the terms in the series in the same form as you would express the first few terms of the descending series, b should be straight forward once you prove the first formula