RE: Random questions

[Martoman]

mmm a few niggling questions I have, some are stupid some I want verified

1) A class of 30 students has a quiz every day. On Monday, 17 students score 100% and the following day 18 score 100%. Find the minimum amount of students who got 100% on both quizzes.

2)On her birthday, 2007, Margret's age is equal to twice the sum of the digits of the year in which she was born. How many possible years are there in which she could have been born? (i know, by inspection 2001 is a solution, but to form a general solution I am having problems)


3) How many 2-digit numbers are equal to three times the product of their digits?

Ok I know for this question that they have to be multiples of 3, as
Then trying all multiples of three I get two, 15,24. This is just time consuming.... any shortcuts or logic I an throw into the mix to make me definitively stop at 24?

4) A rectangular area mesuring 3 units by 4 units on a wall is to be covered with 6 tiles each mesuring 1 unit by 2. In how many ways can this be done?

hmmmmm, i know you can just put the 1-2 tiles in normally as that would cover the area. You can then put them in varying tessellations, but that would take some time, and i KNOW i will miss some possible permutations. So a quick way for this?


5) (pure logic i think, something I seem to be lacking :S) There are four lifts in a building. Each makes three stops, which do not have to be on consecutive floors or include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum number of floors that this building can have?


*edit* 6) number of integer solutions to
I did natural log of both sides,

So
Then null factor , or

In the second case,

Then

So

Trivially

Overall, x = -1,0,3 so three integer solutions? It feels as if it has another one lurking somewhere.

[qshyrn]

first one seems pretty simple:  the minimum is when all that didntget 100 on monday get it on tuesday
so the people thta got 100 on both days would be 18 - (ppl that didnt get 100 on monday)=18-13=5

[brightsky]

Question 6:







(if is even)

But if , is odd, hence discard it.

So we are left with four solutions.

[Martoman]

Question 6:







(if is even)

But if , is odd, hence discard it.



So we are left with four solutions.

Why does that happen if x is even? Is it a property?

*edit* keep on answering in the quote sheesh

[brightsky]

2)On her birthday, 2007, Margret's age is equal to twice the sum of the digits of the year in which she was born. How many possible years are there in which she could have been born? (i know, by inspection 2001 is a solution, but to form a general solution I am having problems)

That sounds like a past AMC problem to me.

By logic, Margret could have been born last century, that is 19xx, or this century 200x.

First consider solutions for 19xx.











Trial and error would leave you with this set of solutions: or or .

Now consider 200x.









Hence there are 4 possible years in which she was born: .

[brightsky]

Question 6:







(if is even)

But if , is odd, hence discard it.



So we are left with four solutions.

Why does that happen if x+1 is even? Is it a property?

*edit* keep on answering in the quote sheesh

You can say it's a property but you can derive it from mere logic.

Say you have:



Now if a is -1, and b is even, because .

[Martoman]

2)On her birthday, 2007, Margret's age is equal to twice the sum of the digits of the year in which she was born. How many possible years are there in which she could have been born? (i know, by inspection 2001 is a solution, but to form a general solution I am having problems)

That sounds like a past AMC problem to me.

By logic, Margret could have been born last century, that is 19xx, or this century 200x.

First consider solutions for 19xx.











Trial and error would leave you with this set of solutions: or or .

Now consider 200x.









Hence there are 4 possible years in which she was born: .

Yes according to the person that sent this to me it was from the junior 07! As if thats junior.

[Martoman]

Question 6:







(if is even)


But if , is odd, hence discard it.



So we are left with four solutions.

Why does that happen if x+1 is even? Is it a property?

*edit* keep on answering in the quote sheesh

You can say it's a property but you can derive it from mere logic.

Say you have:



Now if a is -1, and b is even, because .

*edit*mmmm i didn't type anything here.... i meant to say thanks that makes good sense and why my log thing didn't work.

[TrueTears]

similar q to 6

http://vcenotes.com/forum/index.php/topic,4381.msg56061.html#msg56061

[kamil9876]

5) (pure logic i think, something I seem to be lacking :S) There are four lifts in a building. Each makes three stops, which do not have to be on consecutive floors or include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum number of floors that this building can have?

Best to turn this one into a graph theory (or what you probably call in high school, "networks") problem. The vertices are the floors, the edges are the lifts(and they come in four colours) You want to maximize the vertices which satisfy the condition(any two vertices are joined by one edge).

As a hint: it's obvious that it's not optimal to have any triangles in your network.

edit: read question wrong. Also: edges connecting two floors should indicate that they have a common lift.

This should determin the network easily.

[kamil9876]

3) How many 2-digit numbers are equal to three times the product of their digits?

Ok I know for this question that they have to be multiples of 3, as 3ab = 10a + b
Then trying all multiples of three I get two, 15,24. This is just time consuming.... any shortcuts or logic I an throw into the mix to make me definitively stop at 24?

a(3b-10)=b

a=b/(3b-10)

so now just plug away values of b from 0 to 9. Note that b>3, so start from 4. But also notice that b<3b-10 eventually. (steeper gradient), so once you get that for b=6 that the inequality holds, you can stop since then 0<a<1.