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IKMC 2009 - Grades 9-10 - Self-study

Question 1

Which expression is divisible by 3?
  1. 2009
  2. 2+0+0+9
  3. 200−9
  4. $2^9$
  5. (2+0)(0+9)

Question 2

What is the minimum number of points to remove from the diagram so that no three remaining points are collinear?

Question illustration
  1. 1
  2. 2
  3. 3
  4. 4
  5. 7

Question 3

In a race with 2009 runners, Hans beats three times as many people as finish before him. What is his finishing place?
  1. 503
  2. 501
  3. 500
  4. 1503
  5. 1507

Question 4

Harry delivers a newspaper to every odd-numbered house from 15 through 53 inclusive. How many houses does he visit?
  1. 19
  2. 20
  3. 27
  4. 38
  5. 53

Question 5

Four distinct natural numbers have product 100. What is their sum?
  1. 10
  2. 12
  3. 15
  4. 18
  5. 20

Question 6

Find $\frac12\cdot\frac23\cdot\frac34\cdots\frac9{10}\cdot1000$.
  1. 250
  2. 200
  3. 100
  4. 50
  5. Another number

Question 7

Write 2009 consecutively 2009 times. What is the sum of all odd digits immediately to the left of an even digit?
  1. 2
  2. 9
  3. 4018
  4. 18072
  5. 18081

Question 8

The solid shown has six triangular faces and five vertices. Each vertex has a number; the labels 1 and 5 are shown. The three numbers on every face have the same sum. What is the sum of all five vertex numbers?

Question illustration
  1. 9
  2. 12
  3. 17
  4. 18
  5. 24

Question 9

For how many positive integers a do $a^2$ and $a^3$ have the same number of digits?
  1. 0
  2. 3
  3. 4
  4. 9
  5. Infinitely many

Question 10

The triangle has area 80 m². Each circle has radius 2 m and is centred at a triangle vertex. Find the grey area in m².

Question illustration
  1. 76
  2. $80-2\pi$
  3. $40-4\pi$
  4. $80-\pi$
  5. $78\pi$

Question 11

From its third term onward, each term of a sequence is the sum of the preceding two. The fourth term is 6 and the sixth is 15. What is the seventh?
  1. 9
  2. 16
  3. 21
  4. 22
  5. 24

Question 12

The triangle has one angle 68 degrees and its three angle bisectors are drawn. Find the marked angle between the other two bisectors.

Question illustration
  1. 120°
  2. 124°
  3. 128°
  4. 132°
  5. 136°

Question 13

Maria scores an integer from 0 to 5 on each test. Her average after four tests is exactly 4. Which statement cannot be true?
  1. 4 on every test
  2. 3 exactly twice
  3. 4 exactly twice
  4. 1 exactly once
  5. 3 exactly three times

Question 14

Borromean rings are three rings linked collectively, although no pair is linked: cutting any one lets the other two separate. Which diagram has this property?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 15

Twenty-five nobles and liars stand in a queue. The first says everyone behind him is a liar. Each other person says the person immediately ahead is a liar. Nobles always tell the truth; liars always lie. How many liars are there?
  1. 0
  2. 12
  3. 13
  4. 24
  5. Cannot determine

Question 16

Define $a\star b=ab+a+b$. If $3\star5=2\star x$, find x. The source’s unreadable operator glyph is represented consistently by $\star$.
  1. 3
  2. 6
  3. 7
  4. 10
  5. 12

Question 17

The four circle centres are the vertices of a square. The two large circles touch one another and both small circles. By what factor must a small radius be multiplied to obtain a large radius?

Question illustration
  1. $2/9$
  2. $\sqrt5$
  3. $0.8\pi$
  4. 2.5
  5. $1+\sqrt2$

Question 18

The difference between $\sqrt n$ and 10 is less than 1. How many whole numbers n satisfy this?
  1. 19
  2. 20
  3. 39
  4. 40
  5. 41

Question 19

Friday writes distinct positive integers less than 11 in a row. For every adjacent pair, one number divides the other. What is the maximum possible length of the row?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10

Question 20

Three great circles on a sphere intersect at right angles. Starting at S in the indicated direction, a bug follows them and alternately turns right and left at crossings. How many quarter-circles does it travel before first returning to S?

Question illustration
  1. 6
  2. 9
  3. 12
  4. 15
  5. 18