Một số đề và bài giải Toán OLYMPIAD bậc Tiểu học

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TÔN NỮ BÍCH VÂN
 Suppose today is Tuesday. What day of the week will it be 100 days from
now?
 I have four 30¢ stamps and three 50¢ stamps. Using one or more of these
stamps, how many different amounts of postage can I make?
 Find the sum of the counting numbers from 1 to 25 inclusive. In other
words, if S = 1 + 2 + 3 + ... + 24 + 25, find the value of S.
 In a stationery store, pencils have one price and pens have another price. 2
pencils and 3 pens cost 90¢. But 3 pencils and 2 pens cost 85¢. How much
does 1 pencil cost?
 A work crew of 3 people requires 3 weeks and 2 days to do a certain job.
How long would it take a work crew of 4 people to do the same job if each
person of both crews works at the same rate as each of the others?
Note: Each week contains 6 work days
SOLUTIONS
 Every 7 days from "today" will be Tuesday. Since 98 is a multiple of 7, the 98th day
from today will be Tuesday. Then the 100th day from today will be Thursday.
 Method 1
List the amounts in an organised manner.
Amounts Number
Amounts from 30¢ stamps: 30, 60, 90, 120 4
Amounts from 50¢ stamps: 50, 100, 150 3
Amounts from combining
30¢ stamps and 50¢ stamps: 30+50, 30+100, 30+150 3
60+50, 60+100, 60+150 3
90+50, 90+100, 90+150 3
120+50, 120+100, 120+150 3
Total 19

Method 2
The number of choices we have in using the 30¢ stamps is 5; we can use either 0, 1,
2, 3 or 4 of the 30¢ stamps. Similarly, we have 4 choices with respect to the 50¢ stamps;
we can use either 0, 1, 2 or 3 of the 50¢ stamps. Each of the 5 choices for stamps. This
gives a total of 20¢ combinations. However, this total includes the combination of zero
30¢ stamps and zero 50¢ stamps. Since one or more of the stamps must be used, we
exclude the combination of none of each. Therefore, 19 different amounts of postage can
be made.
 Method 1:
Arrange the numbers like this: 1 + 25 = 25
2 + 23 = 25
3 + 22 = 25
•
•
•
12 + 13 = 25
25 = 25
Method 2:
Given (1) S = 1 + 2 + 3 + ... + 23 + 24 + 25
Reverse order of right side of (1) (2) S = 25 + 24 + 23 + ... + 3 + 2 + 1
Add (1) and (2) (3) 2S = 26 + 26 + 26 + ... + 26 + 26 + 26
Simplify the right side of (3) (4) 2S = 26 x 25
Divide both sides of (4) by 2 (5) S = 13 x 25 or 325
The required sum is 325
 Method 1:
By combining both purchases, we find that 5 pencils and 5 pens cost 175¢. Then 1
pencil and 1 pen cost 35¢, or 2 pencils and 2 pens cost 70¢. Since 3 pencils and 2 pens
cost 85¢, 1 pencil costs 15¢.
Method 2:
Algebra: Let x represent the cost of one pencil and y the cost of one pen.

Give (1) 2x + 3y = 90
Give (2) 3x + 2y = 85
Multiply both sides of (2) by 3 (3) 9x + 6y = 255
Multiply both sides of (1) by 2 (4) 4x + 6y = 180
Subtract (4) from (3) (5) 5x = 75
Divide both sides of (5) by 5 (6) x = 15
Answer: A pencil costs 15¢
 Each person of the work crew of 3 people worked 20 days. Thus the number of
individual work days needed to do the job was 60. Then each member of the work
crew of 4 people must work 15 days in order to provide a total of 60 individual work
days.
13 x 25 = 325