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IKMC 2027 - Grades 11-12 - Self-study SET 4 - Thematic Batch 3

Question 1

How many functions f: R → R satisfy f (x) + 2 f (x + y) = x² + y³ for all real x, y?

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  1. 0
  2. 1
  3. 2
  4. 4
  5. Infinitely many

Question 2

A positive integer N has exactly six distinct integer divisors. The product of five of those six divisors is 648. What is the remaining divisor?

Original printed choices: A) 4; B) 9; C) 12; D) 16; E) 24

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

Question illustration

Question 3

Positive integers x, y satisfy x/43 − y/47 = 1/2021. What is the least possible value of x+y?

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  1. 17
  2. 23
  3. 31
  4. 57
  5. 90

Question 4

If $a^{b}$ =c and $c^{d}$ =5, what is $a^{2 bd}$?

Question illustration
  1. 5
  2. 10
  3. 25
  4. 32
  5. 14

Question 5

Mr Calculator writes a 100-digit positive integer A on the pictured long sheet. An ink stain unfortunately hides the later digits. How many digits does A² have?

Question illustration
  1. 190
  2. 199
  3. 200
  4. 201
  5. Cannot be determined

Question 6

A sprinkler at the centre of a square field waters a circular region tangent to all four sides. Approximately what percentage of the field is not watered?

Question illustration
  1. ≈ 10%
  2. ≈ 20%
  3. ≈ 30%
  4. ≈ 40%
  5. ≈ 50%

Question 7

In Barcelona, pavement tiles have the pattern shown. If the pavement contains leaf shapes, what do those leaves look like?

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  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 8

Three rectangular boxes are joined as shown. One edge is 6 and the indicated face areas are 14,21,16,30. What is the area of the face marked with a question mark?

Question illustration
  1. 18
  2. 24
  3. 28
  4. 30
  5. Cannot be determined

Question 9

Julia places a number in each circle so that the sum at the three vertices of every triangle is the same. What is the largest possible number of distinct values among the seven numbers?

Question illustration
  1. 1
  2. 2
  3. 3
  4. 5
  5. 6

Question 10

A rectangular box has three face areas A, B, C as shown. Which expression equals its volume?

Question illustration
  1. ABC
  2. $\sqrt{ABC}$
  3. $\sqrt{AB+BC+CA}$
  4. $\sqrt[3]{ABC}$
  5. Another answer

Question 11

After New Year celebrations, Alex and Julia play a game with as many beans as the year number: this year they played with 2022 beans. They alternate removing from 1 to 12 beans, and whoever removes the last bean wins. Among future years, which is the earliest year in which the second player has a guaranteed winning strategy?

Question illustration
  1. 2023
  2. 2024
  3. 2026
  4. 2028
  5. The second player cannot win

Question 12

Alice notices that all sign choices for the terms x²,5 x,6 give quadratic equations with integer roots; for example x²+5 x+6=0 has roots −2,−3 and −x²−5 x+6=0 has roots 1,−6. She seeks other expressions x²+bx+c with the same property. Which listed value can c definitely not take?

Question illustration
  1. 24
  2. 30
  3. 48
  4. 54
  5. 60

Question 13

Alex writes two two-digit numbers next to each other. The resulting four-digit number equals three times the product of the original numbers. What is the sum of its digits?

Question illustration
  1. 14
  2. 15
  3. 16
  4. 17
  5. 18

Question 14

In each round of a game Jan gains one ball for a win and loses one otherwise. His chance of winning is 50%. Which value is closest to the probability that he receives 3 balls after 7 rounds?

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  1. 5%
  2. 16%
  3. 2%
  4. 20%
  5. 50%

Question 15

The pictured hexagon has all interior angles equal to 120°. Four consecutive side lengths are 4,18,6,8. What is its perimeter?

Question illustration
  1. 54
  2. 56
  3. 58
  4. 60
  5. 62

Question 16

Albert rolls a six-sided die three times and records the largest result. He repeats this procedure twice more. What is the probability that the three recorded numbers sum to 18?

Question illustration
  1. 1/6⁹
  2. (6⁹ − 5⁹)/6⁹
  3. (6³ − 5³)³/6⁹
  4. 1/2³
  5. 5⁵/6⁹

Question 17

Alex, Bob, Tom race over 1000 m at constant speeds. When Alex finishes, Bob has 100 m left and Tom has 200 m left. When Bob finishes, how many metres does Tom still have to run?

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  1. ≃ 122 m
  2. ≃ 120 m
  3. ≃ 115 m
  4. ≃ 114 m
  5. ≃ 111 m

Question 18

How many two-digit composite numbers are divisible by none of 2,3,5?

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  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 19

A sequence is given by $x_{1}$=2 and $x_{n+1}$=$x_n^{x_n}$ for every positive integer n. What is $x_{4}$?

Question illustration
  1. $2^{2^{3}}$
  2. $2^{2^{4}}$
  3. $2^{2^{11}}$
  4. $2^{2^{16}}$
  5. $2^{2^{768}}$

Question 20

Triangle ABC is inscribed in a circle with centre O and orthocentre H. Points A, O, H are collinear and angle A is 30°. What is angle AHB?

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  1. 125°
  2. 115°
  3. 105°
  4. 95°
  5. 120°