IKMC 2027 - Grades 11-12 - Self-study SET 10 - Thematic Batch 3

Question 1

Alex starts with two apples and one orange in a strange basket. Whenever he takes an apple, one orange disappears and three apples appear. Whenever he takes an orange, one apple disappears and two oranges appear. He takes one fruit per turn, alternating orange, apple, orange, apple, and so on. How many of each fruit remain after 20 turns?

1. 1 apple and 1 orange
2. Only 10 apples
3. 12 apples and 1 orange
4. 20 apples and 10 oranges
5. 22 apples and 11 oranges

Question 2

A cube of side 1 is cut into two equal rectangular boxes. What is the lateral surface area of each box?

Original printed choices: A) 3/2; B) 2; C) 3; D) 4; E) 5

Editorial task: Explain what information is missing or inconsistent, then derive the source’s intended answer under clearly stated additional assumptions.

Question 3

A strip consists of eight consecutive squares, each black or white. A move reverses the colours of four consecutive squares. How many of the four pictured strips can become monochromatic after some moves?

1. 0
2. 1
3. 2
4. 3
5. 4

Question 4

The sum of the positive integers from 1 to n is a three-digit number whose digits are all equal. What is n?

1. 35
2. 36
3. 37
4. 38
5. 39

Question 5

A finite-degree polynomial p(x) with real coefficients satisfies p(x)²=p(p(x))−9 and p(−x)=p(x). What is p(1)?

1. 0
2. 1
3. 3
4. 9
5. None of the other four choices

Question 6

Hugo and Walter take turns rolling an ordinary six-sided die. The first to roll a six wins, and Hugo goes first. What is his probability of winning?

1. 1/4
2. 1/6
3. 1/36
4. 6/11
5. 1/2

Question 7

Question 8

For positive real numbers a, b, define $x_{1}$=a, $y_{1}$=b, $x_{n+1}$=a$\sqrt{y_n}$ and $y_{n+1}$=b$\sqrt{x_n}$. The ratio $x_{5}$/$y_{5}$ can be written as $(a/b)^{m}$. What is m?

1. 5/8
2. 11/16
3. −3/4
4. 21/32
5. −11/8

Question 9

At 1:20, what is the acute angle between the hour and minute hands of a clock?

1. 45°
2. 43°
3. 41°
4. 39°
5. 80°

Question 10

For real x, what is the minimum value of 3x⁴−6x³+3x²+6?

1. 0
2. 3
3. 6
4. −3
5. −6

Question 11

What is the value of Y in the diagram?

1. 6
2. 8
3. 9
4. 10
5. 12

Question 12

Plane mirrors OP and OQ meet at an acute angle. A light ray XY parallel to QO strikes OP at Y, reflects to OQ, then back to OP, and finally travels perpendicular to OQ to R, as shown. If OR=5 cm, what is the distance from XY to OQ?

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

Question 13

Sam reduced a picture’s size by 60%. He now wants another picture whose size is the average of the original and the reduced one. By what percentage should he increase the reduced picture’s size?

1. 40%
2. 45%
3. 50%
4. 60%
5. 75%

Question 14

Square ABCD has side 12. Points E and F lie on BC and CD respectively, with ∠EAB=15° and ∠FAD=30°. What is the area of triangle AEF?

1. 48(3 − $\sqrt{3}$)
2. 12$\sqrt{3}$($\sqrt{3}$ − 1)
3. 48(2$\sqrt{3}$ − 3)
4. 12(3$\sqrt{3}$ − 4)
5. A different result

Question 15

Which listed function has the greatest maximum value on the interval [1,2]?

1. f₁(x) = 1/2 x
2. f₂(x) = 7/2 − 2x²
3. f₃(x) = 5/2 − (x − 1)³
4. 1/8 x⁴
5. 3/2 (x − 1)⁵

Question 16

Let S={1,2,3,…,99,100}. How many three-element subsets K={a, b, c} of S satisfy b=(a+c)/2?

1. 2400
2. 2450
3. 2500
4. 2550
5. 2600

Question 17

What is the greatest possible number of common positive divisors of two arbitrary two-digit positive integers?

Interpretation note: The printed wording does not require distinct integers; the literal answer permits equality. The solution also treats the source’s extra distinctness assumption.

1. 12
2. 11
3. 10
4. 9
5. 8

Question 18

A 3×3×3 cube is built from 27 unit cubes. A move goes to a unit cube sharing a face with the current one, and the central cube may not be entered. How many shortest paths go from one corner cube to the opposite corner cube?

1. 90
2. 72
3. 54
4. 36
5. 18

Question 19

Which expression equals (2015·2016−1)²?

1. 2012 · 2013 · 2014 · 2015 + 1
2. 2013 · 2014 · 2015 · 2016 + 1
3. 2014 · 2015 · 2016 · 2017 + 1
4. 2015 · 2016 · 2017 · 2018 + 1
5. 2016 · 2017 · 2018 · 2019 + 1

Question 20

George colours each region in the pictured design blue, red, purple or yellow. Regions sharing a boundary segment must have different colours. How many colourings are possible?

1. 48
2. 50
3. 64
4. 72
5. 96