1. Multiple Choice Questions (3 points each)
1. What is the answer to 4 × 4 + 4 + 4 + 4 + 4 + 4 × 4 + 4 + 4 + 4 ?
32
48
60
100
144
2. Helen wrote a one-digit number and then wrote an additional digit to its right. She added 19 to the obtained number and got 72. What number did Helen write first?
2
3
5
6
7
3. Jack wishes to write different numbers from 1 to 9 in the cells of the table so that the total of the three numbers of each row and each column were equal to 15. Some of the numbers are already written there. What number must be written in place of the question mark?
Hình ảnh không có chú thích
1
2
3
4
5
4. Each friend: Mary, Elly, Paul, and Mike has exactly one of the following animals: a cat, a dog, a gold fish, and a parrot. Elly has a pet with fur. Mike has a pet with four legs. Paul has a bird, and Mary and Elly don't like cats. Which of the following sentences is not true?
Mike has a dog
Paul has a parrot
Mary has a golden fish
Mike has a cat
Elly has a dog
5. Elise had a sum of money. She gave half of it away to her brother. Then she used half of the remainder to buy a new comic book. Next she gave half of what was left to her older sister. Finally, she found that she had $7 dollars remaining. How much money did she have at first?
30
46
56
65
None of the above
6. Peter has a total of 100 chickens and rabbits on his farm. If the animals have 260 legs altogether, how many chickens are there on the farm?
30
40
50
60
70
7. There are 14 children at a party. If each child shakes hands with each other exactly once, how many handshakes will take place?
75
85
90
91
93
8. The area of the figure in the picture below is equal to:
Hình ảnh không có chú thích
30
38
44
60
None of the above
9. The weight of 3 apples and 2 oranges is 255 g. The weight of 2 apples and 3 oranges is 285 g. Each apple weighs the same and each orange weighs the same. What is the combined weight of 1 apple and 1 orange?
110 g
108 g
105 g
104 g
102 g
10. There were 25 questions in a mathematics quiz. If a question was answered correctly, 4 marks would be given, otherwise 1 marks would be deducted. If John scored 85 marks in the mathematics quiz, how many questions had he answered correctly?
20
21
22
23
24
11. Carl is four times as old as Pam, who is one third as old as Jackie. If Jackie is 18, what is the sum of their ages?
64
54
48
24
18
12. Five cities all got more rain than usual this year. The five cities are: Last Stand, Mile City, New Town, Olliopolis, and Polberg. The cities are located in five different areas of the country: the mountains, the forest, the coast, the desert, and in a valley. The rainfall amounts were: 12 inches, 27 inches, 32 inches, 44 inches, and 65 inches.* The city in the desert got the least rain; the city in the forest got the most rain.* New Town is in the mountains.* Last Stand got more rain than Olliopolis.* Mile City got more rain than Polberg, but less rain than New Town.* Olliopolis got 44 inches of rain.* The city in the mountains got 32 inches of rain; the city on the coast got 27 inches of rain. Which city is in the desert ?
Mile City
Polberg
New Town
Last Stand
Olliopolis
13. If you count 21 letters in the English alphabet from the end and 20 letters from the beginning , which letter will exactly appear in the middle of the sequence thus formed?
K
L
M
N
O
14. A, B, C, D, E, F, G, H and I are nine houses. C is 2km East of B. A is 1km North of B and H is 2km South of A. G is 1km West of H while D is 3km East of G. F is 2km North of G, I is situated just in the middle of B and C, while E is just in the middle of H and D. Find the distance between E and G.
1.5 km
2 km
3 km
4 km
None of the above
15. There is an 8-sided dice whose faces each have one different number from the set {1, 2, 3, 4, 5, 6, 7, 8}. The numbers on opposite faces always have a sum of 9. The dice is rolled twice and the two values appearing on top are multiplied. How many ways are there that this product is divisible by 9?
1
2
3
4
5
Dịch đề: Cho dãy số 2, 7, 12, 17, 22, 27,..., 302, 307. Tìm chữ sồ hàng đơn vị của tích tất cả các số trong dãy.
Solution (in English):
In the sequence, there are \(\left(302-2\right):10+1=31\) numbers that end with 2 and \(\left(307-7\right):10+1=31\) numbers that end with 7.
Therefore, the unit digit of the product is also the unit digit of:
\(A=2\times2\times2\times...\times2\times7\times7\times7\times...\times7\)
(31 numbers 2) (31 numbers 7)
We have to find out what is the final digit of \(2\times2\times...\times2\) (31 times) and \(7\times7\times...\times7\) (31 times)
Consider the number \(N=2\times2\times...\times2\) (\(n\) times)
If \(n=1\), then \(N=2\)
If \(n=2\), then \(N=4\)
If \(n=3\), then \(N=8\)
If \(n=4\), then \(N=16\rightarrow\) the last digit is 6
If \(n=5\), then \(N=32\rightarrow\) the last digit is 2
If \(n=6\), then \(N=64\rightarrow\) the last digit is 4
If \(n=7\), then \(N=128\rightarrow\) the last digit is 8.
If \(n=8\), then \(N=256\rightarrow\) the last digit is 6.
...
From here we can see a pattern: If \(n\) has a remainder of 3 when being divided by 4, the last digit of N is 8. Thus, the final digit of \(2\times2\times...\times2\) (31 times) is 8 since 31 has a remainder of 3 when being divided by 4.
Using the same method of reasoning, we can deduce that \(7\times7\times...\times7\) (31 times) ends with 3.
Therefore, the final digit of A must be the final digit of \(8\times3=24\), which is 4.
So the answer is 4.
The number of terms in the above squence is:
\(\left(307-2\right):5+1=62\) (terms)
The total number squence above is:
\(\left(307+2\right)\times62:2=9579\)
So....