刘以栋 03/24 7089
I did this for my child and just shared here as a reference. I may make mistakes myself.
This is for Math Olympiads, 01/12/2011:
1. When Isaac opens a book, the product of the page numbers on the open pages is 420. Find the sum of the two page numbers.
Solution: Two pages are next to each other, so it is page x and page (x+1), (x+1)*x=420.
x^2+x-420 =0, x=20, x=-21, right answer, x=20
Sum: 20+21 = 41
2 tricks here: two page numbers are close to each other, so page number should be close to the square root of 420. The other trick is that it is the sum of two page numbers. Stop at 20 is wrong.
2. A squirrel buries a total of 80 acorns in N holes. Find the greatest possible of N, provided:
(1) No hole is empty, and
(2) No two holes contain the same number of acorns.
Solution. This number is a little trick, so it is a kind of guessing question. The principal is that, fill the holes from 1, 2, 3 and so on, in that way, you have different numbers in each hole.
The sum of this time series is: x*(x+1)/2, when x=12, the sum is 78. The remaining 2 could not fill one more hole, so just add them to the last hole. N =12.
3. The sum of a proper fraction in lowest terms and its reciprocal equals 2+4/15. Find the original proper fraction.
Solution. x/y+y/x = (x^2+y^2)/xy, 2+4/15 = 34/15, 15 = 3*5, 3^2 + 5^2 =34. The key here is that the fraction is in the lowest form.
The other two questions have pictures and it is a little for me to copy the pictures.