1
geometry
easy
The length of the diagonal of the square is 80 cm. What is the length of a side of the square, in cm? Round off the answer to the nearest integer.
Answer: 57
Side = diagonal / √2 = 80 / √2 = 40√2 ≈ 56.5685, which rounds to 57 cm.
2
algebra
medium
How many perfect squares satisfy the inequality below?
(7 − 3x)/5 ≥ −11
Answer: 5
7 − 3x ≥ −55 ⟹ −3x ≥ −62 ⟹ x ≤ 62/3 ≈ 20.67.
The perfect squares that are ≤ 20.67 are 0, 1, 4, 9 and 16 — 5 perfect squares.
3
algebra
medium
The current ages of Tom and Harry are (2x + 3y) and (5x − 9y), respectively.
Five years ago, the sum of their ages was (ax + by + c). Find the value of (a − b + c).
Answer: 3
Five years ago: (2x + 3y − 5) + (5x − 9y − 5) = 7x − 6y − 10, so a = 7, b = −6, c = −10.
a − b + c = 7 − (−6) + (−10) = 7 + 6 − 10 = 3.
4
algebra
easy
Simplify:
(32.8 × 32.8 + 2 × 32.8 × 67.2 + 67.2 × 67.2) / 10
Answer: 1000
The numerator is a perfect square expansion: 32.8² + 2×32.8×67.2 + 67.2² = (32.8 + 67.2)² = 100² = 10000.
10000 / 10 = 1000.
5
geometry
easy
If an interior angle of a regular polygon with n sides is 150°, how many sides does the polygon have?
Answer: 12
Exterior angle = 180° − 150° = 30°. Number of sides n = 360° / 30° = 12.
6
percentage
medium
Harry bought a bicycle for $450 and he spent $80 on its repair. If he sold the cycle for $600, how much percentage profit did he make? Round off the answer to the nearest integer.
Answer: 13
Total cost = 450 + 80 = $530. Profit = 600 − 530 = $70.
Percentage profit = 70 / 530 × 100% ≈ 13.21%, which rounds to 13%.
7
sequences
easy
Find the 100th term in the sequence below.
2, 5, 8, 11, 14, …
Answer: 299
This is an arithmetic sequence with first term 2 and common difference 3.
100th term = 2 + 99 × 3 = 2 + 297 = 299.
8
measurement
easy
A map is drawn to a scale of 1 : 2700. What is the distance, in cm, between two buildings on the map which are 9450 metres apart?
Answer: 350
9450 m = 945000 cm. Map distance = 945000 / 2700 = 350 cm.
9
rate
medium
Alan can finish a project in 15 days. Ben can finish the same project in 20 days. In how many days can they finish 7/12 of the project if they work on it together?
Answer: 5
Combined rate = 1/15 + 1/20 = 4/60 + 3/60 = 7/60 of the project per day.
Time for 7/12 of the project = (7/12) ÷ (7/60) = (7/12) × (60/7) = 60/12 = 5 days.
10
algebra
medium
It is given that m is inversely proportional to the square of n. It is known that m = 270 for a particular value of n. Find the value of m when this value of n is tripled.
Answer: 30
m = k/n². If n becomes 3n, the new m = k/(3n)² = (k/n²)/9 = 270/9 = 30.
11
geometry
medium
A sphere of radius 4 cm is melted and recast into smaller spheres of radii 2 cm each.
How many such smaller spheres can be made?
Answer: 8
Volume scales with the cube of the radius. Ratio of volumes = (4/2)³ = 8, so 8 smaller spheres can be made.
12
geometry
hard
Segment AB is parallel to segment EF and segment AC is parallel to segment DE. If ∠BAC = 113°, what is the value (in degrees) of ∠DEF?
Answer: 113
Ray ED points in the direction opposite to ray AC, and ray EF points in the direction opposite to ray AB (as drawn in the figure). Reversing both rays that form an angle by 180° leaves the angle between them unchanged, so ∠DEF = ∠BAC = 113°.
13
algebra
medium
The sum of two numbers is 204 and their product is 68. Find the sum of their reciprocals.
Answer: 3
Sum of reciprocals = 1/a + 1/b = (a + b)/(ab) = 204/68 = 3.
14
algebra
medium
Estimate the value of the expression below. Round off the answer to the nearest integer.
(√7 + √3) / (√7 − √3)
Answer: 5
Rationalising: multiply top and bottom by (√7 + √3):
(√7+√3)² / (7−3) = (7 + 2√21 + 3)/4 = (10 + 2√21)/4 = (5+√21)/2.
√21 ≈ 4.583, so the value ≈ (5+4.583)/2 ≈ 4.79, which rounds to 5.
15
algebra
medium
The graph of the curve y = 5x² − kx + 32 intersects the x-axis at x = 4. Find the value of k.
Answer: 28
Substitute x = 4, y = 0: 0 = 5(16) − 4k + 32 = 112 − 4k ⟹ 4k = 112 ⟹ k = 28.
16
algebra
hard
If y = x + 3/z and x = (yz + 1/B) / z, find the value of 1/B².
Answer: 9
From x = (yz + 1/B)/z, we get x = y + 1/(Bz).
From y = x + 3/z, we get x = y − 3/z.
Equating: y − 3/z = y + 1/(Bz) ⟹ −3/z = 1/(Bz) ⟹ −3 = 1/B.
So 1/B² = (−3)² = 9.
17
ratio
medium
If x : y = 7 : 5, find the value of (x³ + y³) / (x³ − y³). Round off the answer to the nearest integer.
Answer: 2
Let x = 7k, y = 5k. x³ = 343k³, y³ = 125k³.
(343 + 125)/(343 − 125) = 468/218 ≈ 2.147, which rounds to 2.
18
statistics
hard
There are some blue, white and orange socks in a drawer. A sock is randomly chosen.
• The probability that the sock is blue or white is 4/7.
• The probability that the sock is white is 9/28.
• The probability that the sock is blue or orange is m/n, where the fraction m/n is in the simplest form.
Find the value of m + n.
Answer: 47
P(blue) = P(blue or white) − P(white) = 4/7 − 9/28 = 16/28 − 9/28 = 7/28 = 1/4.
P(orange) = 1 − P(blue) − P(white) = 1 − 1/4 − 9/28 = 28/28 − 7/28 − 9/28 = 12/28 = 3/7.
P(blue or orange) = 1/4 + 3/7 = 7/28 + 12/28 = 19/28, already in simplest form.
m + n = 19 + 28 = 47.
19
algebra
medium
Given that mn = 4 and m² + n² = 41, find the value of (m + n)².
Answer: 49
(m + n)² = m² + 2mn + n² = 41 + 2(4) = 41 + 8 = 49.
20
speed
medium
A train travelled from station A to station B at a speed of 60 km/h. The train travelled from station B to station A at a speed of 80 km/h. What is the average speed (in km/h) of the train for the whole journey? Round off the answer to the nearest integer.
Answer: 69
For an equal-distance round trip, average speed = 2 × 60 × 80 / (60 + 80) = 9600/140 ≈ 68.57, which rounds to 69 km/h.
21
rate
medium
Two inlet pipes can fill up an empty water tank in 2 hours and 4 hours, respectively. An outlet pipe can empty the same tank filled fully with water in 8 hours. If all the pipes are turned on at the same time, how many minutes will it take to fill up the empty water tank fully?
Answer: 96
Combined rate = 1/2 + 1/4 − 1/8 = 4/8 + 2/8 − 1/8 = 5/8 of the tank per hour.
Time to fill = 1 ÷ (5/8) = 8/5 hours = 1.6 hours = 96 minutes.
22
algebra
hard
If 6/(x²−4) − 1/(x+2) + 1/(x−2) = 2(px+q)/(x²−4), find the value of p² + q².
Answer: 25
Since x² − 4 = (x+2)(x−2): −1/(x+2) = −(x−2)/(x²−4) and 1/(x−2) = (x+2)/(x²−4).
Sum = [6 − (x−2) + (x+2)] / (x²−4) = [6 − x + 2 + x + 2]/(x²−4) = 10/(x²−4).
Matching 10 = 2(px+q) for all x forces p = 0, q = 5.
p² + q² = 0 + 25 = 25.
23
algebra
hard
If 2^(x−5y) = p^(5a) and 2^(2x+7y−238) = p^(10a) where p ≠ 0, find the value of y.
Answer: 14
Squaring the first equation: p^(10a) = (p^(5a))² = 2^(2(x−5y)) = 2^(2x−10y).
Setting this equal to the second equation's exponent: 2x + 7y − 238 = 2x − 10y ⟹ 17y = 238 ⟹ y = 14.
24
number theory
medium
The ratio of two positive integers is 2 : 3 and their Lowest Common Multiple is 174. Find the sum of the two numbers.
Answer: 145
Let the numbers be 2k and 3k. Since gcd(2,3) = 1, LCM(2k, 3k) = 6k.
6k = 174 ⟹ k = 29. The numbers are 58 and 87, summing to 145.
25
algebra
hard
If 1/[(x−y)(y−z)] + 1/[(y−z)(z−x)] + 1/[(z−x)(x−y)] = p − 3, then find the value of p.
Answer: 3
Over the common denominator (x−y)(y−z)(z−x), the numerator is (z−x) + (x−y) + (y−z) = 0, so the whole sum is identically 0.
Therefore p − 3 = 0 ⟹ p = 3.
26
geometry
medium
In right-angled triangle ABC, angle A is the right angle and AD is the height (with D on BC). If AD = 4, find the value of BD × CD.
Answer: 16
In a right triangle, the altitude to the hypotenuse satisfies the geometric mean relation AD² = BD × CD.
BD × CD = 4² = 16.
27
algebra
medium
When (x² − bx + c) is divided by (x + 1), the quotient is (x − 2) and the remainder is −7. Find the value of b − c.
Answer: 10
x² − bx + c = (x+1)(x−2) + (−7) = x² − x − 2 − 7 = x² − x − 9.
Comparing: −b = −1 ⟹ b = 1, and c = −9.
b − c = 1 − (−9) = 10.
28
percentage
easy
The original price of a book is increased by 10% and then decreased by 10%. If the original price of the book is $1300, what is the current price?
Answer: 1287
New price = 1300 × 1.10 × 0.90 = 1300 × 0.99 = $1287.
29
number theory
medium
If the two digits of a two-digit number are reversed, the new 2-digit number is increased by 9. If the sum of the digits is 17, find the original number.
Answer: 89
Let the tens digit be a and units digit be b, with a + b = 17.
Reversing: (10b+a) − (10a+b) = 9 ⟹ 9b − 9a = 9 ⟹ b − a = 1.
Solving a+b=17 and b−a=1 gives a = 8, b = 9. Original number = 89 (check: reversed 98, and 98−89=9).
30
geometry
medium
The area of an equilateral triangle is 25√3 cm². Find the side length of the triangle, in cm.
Answer: 10
Area of equilateral triangle = (√3/4)s². So (√3/4)s² = 25√3 ⟹ s²/4 = 25 ⟹ s² = 100 ⟹ s = 10 cm.
31
geometry
hard
The point P(1,1) is rotated about the origin O in the anti-clockwise direction with OP as the radius by 180°. If R is its new position and the slope of line RO is m = tan(α°), where α° is an acute angle, find the value of (m + α).
Answer: 46
A 180° rotation about the origin sends P(1,1) to R(−1,−1).
Slope of RO = (0 − (−1)) / (0 − (−1)) = 1, so m = 1 and tan(α°) = 1 ⟹ α = 45° (acute).
m + α = 1 + 45 = 46.
32
geometry
medium
The areas of two similar triangles are 90 cm² and 360 cm². The sum of their perimeters is 36 cm. What is the positive difference between their perimeters?
Answer: 12
Ratio of areas = 90:360 = 1:4, so the ratio of corresponding lengths (perimeters) = √1 : √4 = 1:2.
Let the perimeters be p and 2p: p + 2p = 36 ⟹ p = 12, so the perimeters are 12 and 24.
Difference = 24 − 12 = 12.
33
speed
medium
A car travelled a distance of 240 km at a certain uniform speed. If its speed was increased by 20 km/h, it would have taken 1 hour less to cover the same distance. What is the original speed of the car, in km/h?
Answer: 60
Let the original speed be v: 240/v − 240/(v+20) = 1.
240(v+20) − 240v = v(v+20) ⟹ 4800 = v² + 20v ⟹ v² + 20v − 4800 = 0.
Solving: v = [−20 + √(400+19200)]/2 = [−20+140]/2 = 60 km/h.
34
statistics
hard
If the mean of the data set below is 2, find the median of the data.
a² − 22a + 6, 7, 2a − 13, 10a + 18, 17
Answer: 7
Sum of the 5 terms = 5 × 2 = 10.
(a²−22a+6) + 7 + (2a−13) + (10a+18) + 17 = a² − 10a + 35 = 10 ⟹ a² − 10a + 25 = 0 ⟹ (a−5)² = 0 ⟹ a = 5.
Substituting a = 5: terms are −79, 7, −3, 68, 17. Sorted: −79, −3, 7, 17, 68. Median = 7.
35
algebra
hard
(x − 2) is a factor of x² − kx + m and (x − 1) is a factor of x² − kx + m + 5. What is the value of k − m?
Answer: 6
(x−2) a factor: substitute x=2: 4 − 2k + m = 0 ⟹ m = 2k − 4.
(x−1) a factor of x²−kx+m+5: substitute x=1: 1 − k + m + 5 = 0 ⟹ m = k − 6.
Setting equal: 2k − 4 = k − 6 ⟹ k = −2, so m = −2 − 6 = −8.
k − m = −2 − (−8) = 6.