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IKMC 2006 - Grades 11-12 - Self-study

Question 1

Which number is greatest?
  1. 2006×2006
  2. 2005×2007
  3. 2004×2008
  4. 2003×2009
  5. 2002×2010

Question 2

How many trailing zeros does the product of the first 2006 primes have?
  1. 0
  2. 1
  3. 2
  4. 9
  5. 26

Question 3

How many more squares can be coloured grey to increase the grey area without increasing its perimeter?

Question illustration
  1. 0
  2. 7
  3. 18
  4. 12
  5. 16

Question 4

Each of the four cards has a letter on one side and a number on the other. Raza claims every vowel has an even number on its reverse. What is the fewest cards Sara must turn to check this?

Question illustration
  1. none
  2. 1
  3. 2
  4. 3
  5. 4

Question 5

Two equal-length trains travel oppositely at 100 and 120 km/h. A passenger on the second sees the first pass completely in 6 seconds. How long does a passenger on the first see the second pass?
  1. 5 sec
  2. 6 sec
  3. between 6 and 7 sec
  4. 7 sec
  5. more than 7 sec

Question 6

Two pendants have the same material, thickness and weight. One is an annulus of concentric radii 6 cm and 4 cm, the other a solid disk. Find its radius.

Question illustration
  1. $4$ cm
  2. $2\sqrt6$ cm
  3. $5$ cm
  4. $2\sqrt5$ cm
  5. $\sqrt{10}$ cm

Question 7

The consecutive differences in a, b, c, d, e are equal. If b=5.5, e=10, find a.
  1. 0.5
  2. 3
  3. 4
  4. 4.5
  5. 5

Question 8

If $4^x=9$ and $9^y=256$, find xy.
  1. 2
  2. 4
  3. 10
  4. 36
  5. 48

Question 9

Every nine-digit integer using digits 1, …, 9 once is put on a separate sheet in a box. How many sheets guarantee at least two numbers with the same first digit?
  1. 9!
  2. 8!
  3. 72
  4. 10
  5. 9

Question 10

$AB=1$, $\angle ABC=\angle ACD=90^\circ$, $\angle CAB=\angle DAC=\theta$. Find AD.

Question illustration
  1. $\cos\theta+\tan\theta$
  2. $1/\cos2\theta$
  3. $\cos^2\theta$
  4. $\cos2\theta$
  5. $1/\cos^2\theta$

Question 11

Which function has a graph symmetric about the y-axis?
  1. $y=x^2+x$
  2. $y=x^2\sin x$
  3. $y=x\cos x$
  4. $y=x\sin x$
  5. $y=x^3$

Question 12

A fair roulette wheel has 0 and 1, …, 36. What is the probability of a prime outcome?
  1. 5/18
  2. 11/37
  3. 11/36
  4. 12/37
  5. 1/3

Question 13

Dividing 1001 by a one-digit number leaves remainder 5. What remainder does 2006 leave with that divisor?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 14

A circular traffic sign has radius 20 cm. Each dark piece is a quarter-circle; the four dark pieces together equal the light area. What is the radius of a dark quarter-circle?

Question illustration
  1. $10\sqrt2$ cm
  2. $4\sqrt5$ cm
  3. $20/3$ cm
  4. $12.5$ cm
  5. $10$ cm

Question 15

Primes a>b>c satisfy $a+b+c=78$, $a-b-c=40$. Find abc.
  1. 438
  2. 590
  3. 1062
  4. 1239
  5. 2006

Question 16

The sector radius and its incircle radius are in ratio 3: 1. Find sector-area: incircle-area.

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

Question 17

Sixteen volleyball teams each play every other once. A win gives 1 point, a loss 0, no draws. The final scores form an arithmetic sequence. How many points does the last team have?
  1. 3
  2. 2
  3. 1
  4. 0
  5. none of these

Question 18

Last year a singing team had 30 more boys than girls. This year total membership rose 10%, girls 20%, boys 5%. How many members this year?
  1. 88
  2. 99
  3. 110
  4. 121
  5. 132

Question 19

A move swaps two cells in the same row or column. What minimum number transforms Fig.1 into Fig.2?

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

Question 20

The window uses red R, green G, blue B glass as shown. Green glass totals 400 cm². How much blue is required?

Question illustration
  1. $396$
  2. $400$
  3. $120\pi$
  4. $90\sqrt2\pi$
  5. $382$