Math Olympiad Essay Solutions

Solution 1 :
x = 1/6x + 20 + 55
5/6x = 75
x = 90

Solution 2 :
c + d = 335, c = 2/3d
2/3d + d = 335
5/3d = 335
d = 201

Solution 3 :
M = 120 : 2 = 60 : 2 = 30
B  = 144 : 2 = 72 : 2 = 36
P  = 100 : 2 = 50 : 2 = 25
Max 2 x 2 = 4 schools.

Solution 4 :
60 x t = 45 (t + 4/3)
60t = 45t + 60
15t = 60
t = 4 hours

Solution 5 :

Solution 6 :
15 x t = 12 (t + 1/4)
15t = 12t + 3
3t = 3
t = 1 hour
s = 15 km

Solution 7 :
Average A = (140 + 125 + 135 + 141 + 135 + 142 + 123) / 7 = 134.43 days
Average B = (135 + 125 + 145 + 122 + 139) / 5 = 133.20 days
Bulb A is longer then B

Solution 8 :
5739
5793
3579
9573
5379
5973

Solution 9 :
1 - 9 = 9 digit
10 - 99 = 10 x 9 x 2 = 180 digit, total 189 digit
100 - 199 = 10 x 10 x 3 = 300 digit, total 489 digit
200 - 299 = 10 x 10 x 3 = 300 digit, total 789 digit, (840 - 789) / 3 = 51 / 3 = 17
300 - 316 = 17 x 3 = 51 digit, total 840 digit.
Total 316 pages.

Solution 10 :
Number should be odd, because sum of two consecutive number always odd.
Odd number less then 2005 is 2003.
Least odd number is 2 + 3 + 4 = 4 + 5 = 9
Odd number 2003 to 9 = 998 numbers