Section A
Multiple Choice Questions — 15 Questions
(+4 / -1)
1
arithmetic
easy
What is the value of (115 + 114 + 113 + 112 + 111) − (101 + 102 + 103 + 104 + 105)?
The source PDF prints no A/B/C/D/E letters for these options; labels follow the exact on-page top-to-bottom order.
A
50
B
60
C
70
D
80
E
100
Show answer
Answer: A
The difference between the first term in the first group and the last term in the second group is 115 − 105 = 10.
The difference between the second term of the first group and the second last term of the second group is also 114 − 104 = 10.
This pattern continues for each of the rest of the term pairs. There are five pairs whose difference is 10, therefore the value is 5 × 10 = 50.
2
number-theory
medium
How many digits does N have if N = 2 × 3 × 4 × 5 × 12 × 25?
A
3
B
4
C
5
D
6
E
7
Show answer
Answer: C
Reorder and insert parentheses to show grouping: (2×5) × (4×25) × (3×12) = 10 × 100 × 36 = 36000.
N has 5 digits.
3
arithmetic
medium
Find the value of 10×1 − 9×2 + 8×3 − 7×4 + 6×5 − 5×6 + 4×7 − 3×8 + 2×9 − 1×10.
A
0
B
1
C
10
D
50
E
100
Show answer
Answer: A
Notice that 10×1 = 1×10, 9×2 = 2×9, 8×3 = 3×8, 7×4 = 4×7 and 6×5 = 5×6. Thus all 10 products have a sum of 0.
4
number-theory
medium
The product of the digits of the number 217 is 14. What is the least 3-digit odd number whose product of the digits is 6?
A
123
B
132
C
146
D
164
E
183
Show answer
Answer: A
Consider the hundreds digit to be 1. The product of the last two digits has to be 6.
Since 2 × 3 = 6, the least 3-digit odd number must be 123.
5
algebra
medium
Last holiday, Jonathan received $134 more than three times the money that Sally received. They received $710 altogether. How much money did Jonathan receive?
A
$144
B
$432
C
$566
D
$576
E
$288
Show answer
Answer: C
Draw a model: Sally = 1 unit, Jonathan = 1 unit + 1 unit + 1 unit + $134, Total = 710.
4 units + 134 = 710.
1 unit = (710 − 134) ÷ 4 = 144.
Jonathan received 144 × 3 + 134 = 566.
6
fractions
hard
Amy spent 1/4 of the cash in her wallet at the game store, 2/3 of the remaining cash at a clothing store, 1/3 of what was left for lunch, and 3/4 of what remained at the shoe store. After spending all that money, she had $10 left. How much cash did she have at first?
A
$24
B
$120
C
$160
D
$240
Show answer
Answer: D
The word "of" means multiply in this case.
After the game store, Amy left 3/4 of the cash in her wallet.
After the clothing store, she left 3/4 × 1/3 = 1/4 of the cash in her wallet.
After lunch, she left 1/4 × 2/3 = 1/6 of the cash in her wallet.
After the shoe store, she left 1/6 × 1/4 = 1/24 of the cash in her wallet.
1/24 of the cash in her wallet is equal to $10. So, Amy had $10 × 24 = $240 at first.
7
algebra
medium
The sum of three consecutive whole numbers is 443 more than the greatest of them. What is the least of the three numbers?
A
219
B
220
C
221
D
222
E
442
Show answer
Answer: C
Let the smallest of these three consecutive numbers be 1 unit.
Then the second and third numbers are (1 unit + 1) and (1 unit + 2), respectively.
It is given that 1 unit + (1 unit + 1) + (1 unit + 2) = (1 unit + 2) + 443.
3 units + 3 = 1 unit + 445.
3 units − 1 unit = 445 − 3.
2 units = 442.
1 unit = 221.
8
number-patterns
medium
What is the 100th number in the sequence below?
17, 24, 31, 38, 45, 52, ...
A
703
B
710
C
717
D
724
E
731
Show answer
Answer: B
Notice that
1st number: 7×1 + 10 = 17
2nd number: 7×2 + 10 = 24
3rd number: 7×3 + 10 = 31 and so on.
Thus, the 100th number is 7×100 + 10 = 710.
9
logic-puzzle
hard
In a magic square, the sum of the numbers in each row, column and diagonal are equal. The odd numbers 1 through 17 are placed in the magic square shown. What number goes in the square marked with "A"?
A
7
B
9
C
11
D
15
E
17
Show answer
Answer: A
The sum of the numbers in the magic square is 1+3+5+7+9+11+13+15+17 = 81. Hence the sum of the numbers in each row, column and diagonal is 81 ÷ 3 = 27.
The number in the bottom right square is 27 − 13 − 3 = 11 and the number in the middle square is 27 − 5 − 13 = 9. Thus, the number A in the top left square is 27 − 11 − 9 = 7.
10
algebra
medium
If 52 is added to two-fifths of a number, the triple of the number is the result. What is the number?
A
8
B
12
C
16
D
20
E
24
Show answer
Answer: D
Let the number be 5 units. Then 52 + 2/5 of 5 units = 3 × 5 units.
52 + 2 units = 15 units.
52 = 15 units − 2 units = 13 units.
1 unit = 52 ÷ 13 = 4.
Hence the number is 5 × 4 = 20.
11
algebra
medium
Given that ▲ + ■ + ◆ = 14, ■ + ◆ + ● = 19, ◆ + ● + ▲ = 16 and ● + ▲ + ■ = 11, what is the value of ▲?
A
1
B
4
C
6
D
9
E
20
Show answer
Answer: A
Sum up all the equations.
(▲+■+◆) + (■+◆+●) + (◆+●+▲) + (●+▲+■) = 14+19+16+11 = 60.
3 × (▲+■+◆+●) = 60 → (▲+■+◆+●) = 20.
So, ▲ = 20−19 = 1, ■ = 20−16 = 4, ◆ = 20−11 = 9 and ● = 20−14 = 6.
The value of ▲ is 1.
12
number-theory
medium
What is the positive difference between the sum of the multiples of 3 less than 100, and the sum of the multiples of 4 less than 100?
A
483
B
1200
C
1683
D
3366
E
2400
Show answer
Answer: A
The multiples of 3 are 3, 6, 9, ..., 93, 96 and 99. There are 33 such numbers. The sum of the multiples of 3 is (3+99) × 33 ÷ 2 = 1683.
The multiples of 4 are 4, 8, 12, ..., 92 and 96. There are 24 such numbers. The sum of the multiples of 4 is (4+96) × 24 ÷ 2 = 1200.
So, the difference between the two numbers is 1683 − 1200 = 483.
13
volume
hard
The cube at the right is constructed of the same-sized figures as shown. How many figures does the cube contain?
A
18
B
19
C
20
D
21
E
22
Show answer
Answer: A
Based on the diagram, the dimension of each figure (1 slab) is 1 unit × 2 units × 6 units. Thus, the volume of each figure is 12 units³.
The volume of the cube is 6×6×6 = 216 units³.
Therefore, the number of figures is 216 ÷ 12 = 18.
14
algebra
medium
Gina has $1350 and receives $11 every week. James has $1650 and spends $19 per week. After how many weeks will they have the same amount of money?
A
7
B
8
C
9
D
10
E
11
Show answer
Answer: D
Since the difference between receiving $11 and spending $19 is $30, the difference between Gina and James's dollar amounts shrinks by $30 each week. James has $1650 and Gina has $1350 at the start, so the difference in their starting amounts is 1650 − 1350 = $300. At $30 per week, it will take 300 ÷ 30 = 10 weeks for them to have the same amount of money.
15
algebra
hard
Given □ + Δ + ◇ = 30, Δ + ◇ + ★ = 22, ◇ + ★ + □ = 26, and Δ + ◇ + ★ + □ = 37, find the value of ★ + □ + Δ.
A
7
B
11
C
15
D
33
E
26
Show answer
Answer: D
Since Δ + ◇ + ★ + □ = 37 and:
• Δ + ◇ + ★ = 22, 22 + □ = 37 so □ = 15
• □ + Δ + ◇ = 30, 30 + ★ = 37 so ★ = 7
• ◇ + ★ + □ = 26, 26 + Δ = 37 so Δ = 11
Thus, ★ + □ + Δ = 7 + 15 + 11 = 33.
Section B
Open-Ended (Integer) Questions — 10 Questions
(+5)
16
number-theory
medium
Jean has a collection of chips. If the chips are arranged in piles of 12, there will be 11 chips left over. If the chips are arranged in piles of 21, there will be 20 chips left over. What is the least number of chips that Jean owns?
Show answer
Answer: 83
If we add 1 to the total number of chips, then the new number N is divisible by 12 and 21. The smallest possible value of N is the least common multiple of 12 and 21 which is 84. Hence the least number of chips that Jean owns is 84 − 1 = 83.
17
cryptarithm
hard
In the following cryptarithm, each different letter represents a different digit. Find the last three digits of the largest possible sum. (For example, if your answer is 1234, then write 234.)
MINED
+ DENIM
Show answer
Answer: 147
Since M and D have the highest place values, let them represent 8 and 9. Since I and E have the second highest place values, let them represent 6 and 7. Since N has the next highest place value, let it represent 5. Note that M and D share the same place value, therefore each letter could represent either digit (8 or 9) and would yield the same sum. The same applies to I and E. Therefore, the last three digits of the largest possible sum [sample representation: 97568 + 86579] are 147.
18
area
hard
Four shaded right-angled triangles with leg lengths 1 cm × 4 cm, 2 cm × 4 cm, 3 cm × 4 cm, and 4 cm × 4 cm are fitted together to make a quadrilateral. This quadrilateral is then fit into a 5 cm × 8 cm rectangle. Find the area, in cm², of the quadrilateral that was formed (the shaded region). (Legs of a right-angled triangle are the sides beside the right angle.)
Show answer
Answer: 20
A diagonal cuts a rectangle into exactly two equal parts. Notice that the quadrilateral is formed by the four shaded triangles and each of the shaded triangles is exactly half of the smaller rectangles. The area of the 5×8 rectangle is 40 cm² and we know that the area of the smaller quadrilateral is half of the original rectangle. Half of 40 is 20 cm².
19
rate-work
medium
Adam can paint a classroom in 6 hours. Ben can paint the same classroom in 12 hours. If they work together, each at their own rate, to paint this classroom, how many hours would they take?
Show answer
Answer: 4
Adam can paint 1/6 of a classroom in one hour. Ben can paint 1/12 of a classroom in one hour. Together, they can paint 1/6 + 1/12 = 1/4 of a classroom in one hour. This means to paint the entire classroom, they need 4 × 1 = 4 hours.
20
combinatorics
hard
Each of the whole numbers 1 through 9 is written on 9 cards, one number per card. Three cards are dealt to each of three players. The sum of Player One's cards is 15. The sum of Player Two's cards is 23. What is the highest single card Player Three could have?
Show answer
Answer: 4
There is only 1 way to get 23 as the sum of 3 numbers: 9 + 8 + 6 = 23.
Therefore, Player Two gets 6, 8 and 9.
Afterwards, there is 1 way to get 15 as the sum of 3 numbers: 7 + 5 + 3 = 15.
Therefore, the highest single card Player Three could have is 4.
21
combinatorics
hard
Orville has L-tiles, each made up of three 1×1 squares (as shown). What is the greatest number of these L-tiles that can fit into a 14×10 rectangle so long as there is no overlap? (Note: The L-tiles can be rotated in order to fit better but not cut.)
Show answer
Answer: 46
Create 2×3 blocks and pack them in as tightly as possible.
A good guess for the answer can be found by dividing the number of 1×1 squares in the large rectangle by the number of 1×1 squares in the L-tile. Notice that 140 ÷ 3 = 46 R 2, so the answer cannot be greater than 46. Can the answer be 46? Yes, if the tiles are packed as shown in the diagram (with the two shaded squares left empty). Therefore, the greatest number of these L-tiles that can fit into a 14×10 rectangle is 46.
22
logic-puzzle
hard
The numbers 1 through 12 are placed in the diagram, one in each circle, so that the sum of the numbers along each line is the same. What is the smallest possible value of this sum per line?
Show answer
Answer: 28
Let the sum of the numbers along each line be S and the numbers on the vertices (circles) of the triangle be a, b and c.
Next, let us add all the numbers along the 3 lines. We can notice that the numbers 1 through 12 except a, b and c were added only once whereas a, b and c were added twice. Thus,
3S = (1+2+3+...+12) + a+b+c = 78 + a+b+c.
S = (78 + a+b+c) / 3.
The smallest possible value of S is obtained when a+b+c is the smallest possible.
Hence S must be (78+1+2+3)/3 = 84/3 = 28.
23
number-theory
hard
How many perfect squares less than or equal to 200 are the sum of two consecutive whole numbers?
Show answer
Answer: 7
Two consecutive whole numbers must consist of one odd and one even integer. Hence, the sum of two consecutive integers is always an odd number. Since the square of an even number is always even and the square of an odd number is always odd, make a list of the squares of consecutive odd integers starting with 1:
1² = 1 = 0+1
3² = 9 = 4+5
5² = 25 = 12+13
7² = 49 = 24+25
9² = 81 = 40+41
11² = 121 = 60+61
13² = 169 = 84+85
Odd numbers squared greater than 13² result in a number greater than 200. Therefore, there are only 7 perfect squares less than 200 equal to the sum of two consecutive integers.
24
rate-work
medium
Karen can paint a wall in 2.5 hours. Stephanie takes 1.25 hours to paint the same size wall. How many minutes will it take for Karen and Stephanie working together to paint the same wall?
Show answer
Answer: 50
Since 2.5 = 2 1/2 = 5/2 and it takes Karen 5/2 hours to paint the wall, she completes 2/5 of the wall in 1 hour. Similarly, since 1.25 = 1 1/4 = 5/4, and it takes Stephanie 1 1/4 hours to paint the wall, she completes 4/5 of the wall in 1 hour. Working together they complete 2/5 + 4/5 = 6/5 of the wall in 1 hour. They only need to complete 1 wall, so it takes them 5/6 hours or 5/6 (60) = 50 minutes.
25
algebra
medium
The cost of 3 pencils and 1 marker is $2.10. The cost of 1 pencil and 5 markers is $6.30. What is the total cost of 14 pencils and 14 markers?
Show answer
Answer: 21
Let P = cost of a pencil and let M = cost of a marker.
Then 3P + M = 2.10 and P + 5M = 6.30. Adding the first equation 2 times to the second equation will get 7P + 7M = 10.50. Hence, the total cost of 14 pencils and 14 markers is 2 × $10.50 = $21.