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IKMC 2027 - Grades 9-10 - Self-study SET 3 - Thematic Batch 3

Question 1

Triangle ABC is isosceles with apex A. Point P lies on the bisector of ∠BAC, as shown, and ∠BPC=2∠PBC. What is ∠CAP+∠ACP?

Question illustration
  1. 135°
  2. 80°
  3. 50°
  4. 45°
  5. 65°

Question 2

The three-digit number $\overline{ab0}$ equals the sum of the cubes of its digits. Which of the following numbers also has this property?

Question illustration
  1. $\overline{a0b}$
  2. $\overline{(a−1)b0}$
  3. $\overline{a(b−1)0}$
  4. $\overline{ab1}$
  5. $\overline{a1b}$

Question 3

Alex, Bob and Tom race over 1000 m. When Alex finishes, Bob is 100 m from the finish and Tom is 200 m away. All three run at constant speeds. How far is Tom from the finish when Bob finishes?

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

Question 4

Numbers a, b, c, d satisfy $a+5=b^2-1=c^2+3=d-4$. Which is the greatest?

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

Question 5

Bob and Tom run around a closed track, taking 4 and 6 minutes respectively for one lap. They start together at the same point in opposite directions. How long until they next meet?

Question illustration
  1. 2 minutes
  2. 2 minutes 20 seconds
  3. 2 minutes 24 seconds
  4. 2 minutes 30 seconds
  5. 3 minutes

Question 6

A cube is assembled from equal unit cubes, each entirely black or entirely white. The figures show the cube from two different directions. What is the greatest possible number of black unit cubes?

Question illustration
  1. 5
  2. 7
  3. 8
  4. 9
  5. 10

Question 7

What is the probability that it will rain today?

Editorial note: Explain which information is missing and whether a numerical probability follows.

Question illustration

Question 8

Let $M=2016\times5^{2016}$. What is the greatest number of trailing zeros in its decimal representation?

Question illustration
  1. 1
  2. 2
  3. 5
  4. 32
  5. 2016

Question 9

Tom has three times as many sweets as Sam. Tom gives Sam half of his own sweets, so Sam now has more than Tom. What percentage of his current sweets must Sam give back so they have equal amounts?

Question illustration
  1. 10%
  2. 20%
  3. 25%
  4. 40%
  5. 50%

Question 10

Julia colours some cells of a 4×4 grid grey so that every white cell shares a side with at least two grey cells. What is the smallest possible number of grey cells?

Question illustration
  1. 5
  2. 6
  3. 7
  4. 8
  5. 9

Question 11

Two squares are respectively inscribed in and circumscribed about two equal circles. What is the ratio of the smaller square’s perimeter to the larger square’s perimeter?

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

Question 12

Julia fills every square box in the diagram with a number so that the three numbers on each diameter sum to 13 and the numbers on the circle sum to 40. What should she put at the centre?

Question illustration
  1. 12
  2. 3
  3. 10
  4. 8
  5. 5

Question 13

A manager makes two statements about the pairwise distinct salaries of Francesc, Maria and Joana. If Francesc is not paid the most, Joana must be. If Joana is not paid the least, Maria is paid the most. Who is paid the most and who the least, respectively?

Question illustration
  1. Maria. Joana
  2. Francesc. Joana
  3. Joana. Maria
  4. Maria. Francesc
  5. Francesc. Maria

Question 14

A postal worker puts three letters into 10 boxes in a row, with at most one letter per box and no two occupied boxes consecutive. How many placements are possible?

Question illustration
  1. 9
  2. 15
  3. 24
  4. 35
  5. 56

Question 15

Eighty-two people compete in darts. Each receives a positive integer score at most 9. The greatest number of people who can be selected with equal scores is n. What is the sum of the largest and smallest possible values of n?

Question illustration
  1. 83
  2. 85
  3. 90
  4. 92
  5. 102

Question 16

The diagram contains three adjacent figures: a rectangle, an equilateral triangle and a regular pentagon. Each has two vertices on the circle. Find the angle x.

Question illustration
  1. 30°
  2. 36°
  3. 40°
  4. 43°
  5. 44°

Question 17

Julia fills each of the seven circles shown with a number so that the sums at the three vertices of every drawn triangle are equal. What is the greatest number of distinct values among the seven entries?

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

Question 18

Three people walk from A to B at 4,5 and 6 km/h. The distance is very large. The second starts 15 minutes after the first. How many minutes after the second should the third start so that all three meet somewhere along the route?

Question illustration
  1. 3 minutes
  2. 5 minutes
  3. 10 minutes
  4. 25 minutes
  5. The three cannot meet

Question 19

Last week Anna paid 2.2 dollars for 1 kg of apples and 1 kg of oranges. Today the same amounts cost 2.6 dollars. Orange prices have risen 15% and apple prices 20%. What is the current price of 1 kg of apples?

Question illustration
  1. 1.4 dollars
  2. 1.48 dollars
  3. 1.56 dollars
  4. 1.68 dollars
  5. Cannot be determined

Question 20

Katie and Petra use a faulty scale whose reading differs from the true weight by a fixed amount. Separately they read 52 kg and 54 kg. Together the reading is 111 kg. What is Petra’s true weight?

Question illustration
  1. 47 kg
  2. 49 kg
  3. 51 kg
  4. 57 kg
  5. 59 kg