IKMC 2027 - Grades 11-12 - Self-study SET 2 - Thematic Batch 3

Question 1

A square is inscribed in triangle ABC as shown. The square divides the triangle’s midline into three segments AP, PQ and QM satisfying AP/1=PQ/2=QM/2. What is the ratio of the area of triangle ABC to that of the square?

1. 25/16
2. 16/9
3. 25/8
4. 16/6
5. 5/2

Question 2

Let a and b be perfect squares such that a−b is prime. Which value below can neither of the two squares have?

1. 100
2. 144
3. 400
4. 625
5. 2500

Question 3

A sequence begins with $a_{1}$=$a_{2}$=1. Its later terms satisfy $a_{n+2}$=$a_{n}$+$(−1)^{n}$ $a_{n+1}$. For example, $a_{3}$=$a_{1}$−$a_{2}$, $a_{4}$=$a_{2}$+$a_{3}$, $a_{5}$=$a_{3}$−$a_{4}$, and so on. What is the sum of its first 100 terms?

1. 0
2. 3
3. −21
4. 100
5. −1

Question 4

In the diagram, the grey regions are bounded by a unit square and the parabolas y=x (2−x) and y=(x−1)². The region between y=x (2−x) and the horizontal axis has area 4/3. What is P₁−P₂?

1. 1/3
2. 2/3
3. 1/4
4. 1/2
5. Cannot be determined

Question 5

At 9:40 robot R 1 leaves a laboratory, travelling due east along a very long straight road at 25 km/h. At 10:04 robot R 2 leaves the same laboratory, travelling due south along a very long straight road at 37.5 km/h. At what time is their straight-line distance exactly 150 km?

1. 12:40
2. 12:45
3. 12:50
4. 13:04
5. 13:16

Question 6

Two natural numbers have greatest common divisor 6 and least common multiple 900. Which of the following cannot be their sum?

1. 318
2. 270
3. 186
4. 462
5. 906

Question 7

Consider equation (E): x³−5 x+a=0, where a is real. Which assertion below is certainly true?

1. Equation(E) has at most one integer root for every real a.
2. There exists a real a for which(E) has two integer roots.
3. There exists a real a for which(E) has three integer roots.
4. For every real a, equation(E) has no integer root.
5. For every real a, equation(E) has only irrational roots.

Question 8

Positive integers a, b, c, d satisfy a+2=b−2=2 c=d÷ 2. Which of the four is largest?

1. a
2. b
3. c
4. d
5. Cannot be determined

Question 9

The positive integers 1 through 20 are to be divided into ten pairs, with the same difference between the two numbers in every pair. Order within each pair is irrelevant. How many such divisions are possible?

1. 0
2. 3
3. 4
4. 5
5. More than five divisions meet the requirements.

Question 10

In the diagram, PQRS, TUVR and UWXY are squares, and P, T, X are collinear. The first two squares have areas 36 and 16 respectively. What is the area of triangle PXV?

1. 14 2/3
2. 15 1/3
3. 16
4. 17 2/3
5. 18

Question 11

There are many wooden sticks of each length 6,5,4,3,2,1 cm. By selecting three sticks, how many different triangle sizes can be formed?

1. 12
2. 18
3. 24
4. 30
5. 34

Question 12

A piece of string lies on a table with parts hidden as shown. At each hidden crossing it can have either of the two pictured over-under arrangements and . If its ends are pulled, what is the probability that it has a knot?

Original printed choices: A) 1/2; B) 1/4; C) 1/8; D) 3/4; E) 3/8

Editorial note: Explain whether the printed question and choices are consistent. Give the literal mathematical result and distinguish any intended correction.

Question 13

A function f: R→R satisfies f (x)≥0 for all real x, f (1)=2017, and f (x+y)=f (x)f (y) for all real x, y. What is f (1/2)?

1. 2017/2
2. 1
3. 0
4. 4034
5. √2017

Question 14

A straight line through X bisects the area of the grey region. Through which of the points A, B, C, D, E does it pass?

1. A
2. B
3. C
4. D
5. E

Question 15

Tom draws a square and joins each vertex to the midpoints of the two nonadjacent sides. This produces the inner octagon shown. What is the ratio of the octagon’s area to the square’s area?

1. 1/6
2. 3/14
3. 2/11
4. 1/5
5. 1/8

Question 16

In the diagram, the smallest square shares a vertex with each of the other two squares. The numbers inside two squares are their respective areas. What is the largest square’s area?

1. 60
2. 70
3. 80
4. 90
5. 100

Question 17

A triangle has side lengths a≤5 cm, b≤6 cm and c≤10 cm. What is the greatest possible length of an altitude, in centimetres?

1. 5,5 cm
2. 6 cm
3. 6,5 cm
4. 8 cm
5. 9,5 cm

Question 18

A four-digit number has no digit 0 or 9. Decrease its first and last digits by 1 and increase each middle digit by 1 to form a new number. The greatest common divisor of the old and new numbers is the two-digit number ab, greater than 85. What is a+b?

1. 10
2. 12
3. 15
4. 17
5. 18

Question 19

A number a satisfies a²−a/2=1/4. What is a³−a/2?

1. −1/4
2. 1/4
3. 1/2
4. 4
5. 1/8

Question 20

The graph of f (x)=x²+bx+c is tangent to the horizontal axis at (r,0). Which of the following must equal b?

Original printed choices: A) r; B) 2r; C) ±√c; D) bc/r; E) c/r

Editorial note: Explain whether the printed question and choices are consistent. Give the literal mathematical result and distinguish any intended correction.