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IKMC 2014 - Grades 11-12 - Self-study

Question 1

Remove some unit cubes from a 5×5×5 cube to obtain the shown solid: equally high pillars on a common base. How many cubes were removed?

Question illustration
  1. 56
  2. 60
  3. 64
  4. 68
  5. 80

Question 2

Today is Carmen, Gerda and Sabine’s birthday, and their ages sum to 44. What will the sum be the next time it is a two-digit number with equal digits?
  1. 55
  2. 66
  3. 77
  4. 88
  5. 99

Question 3

Find $a^{-3k}$ if $a^k=1/2$.
  1. $1/8$
  2. $8$
  3. $-8$
  4. $6$
  5. $1/6$

Question 4

Three baskets of different sizes contain 48 balls altogether. The smallest and largest together contain twice as many as the middle basket. The smallest contains half as many as the middle. How many are in the largest?
  1. 16
  2. 20
  3. 24
  4. 30
  5. 32

Question 5

Calculate $\dfrac{2^{2014}-2^{2013}}{2^{2013}-2^{2012}}$.
  1. $2^{2011}$
  2. $2^{2012}$
  3. $2^{2013}$
  4. $1$
  5. $2$

Question 6

Which expression does not have b+1 as a factor?
  1. $2b+2$
  2. $b^2-1$
  3. $b^2+b$
  4. $-1-b$
  5. $b^2+1$

Question 7

How many digits does $(2^{22})^5(5^{55})^2$ have?
  1. 22
  2. 55
  3. 77
  4. 110
  5. 111

Question 8

Handsome Fritz has a secret email address known only to four friends. Today he received eight emails there. Which statement must be true?
  1. Two emails came from each friend.
  2. All eight cannot come from one friend.
  3. At least one came from every friend.
  4. At least two came from one friend.
  5. At least two came from each of at least two friends.

Question 9

The lateral surfaces of two identical cylinders are cut along vertical lines and joined side by side, as shown, to form one larger cylinder’s lateral surface. How does its volume compare with one original cylinder?

Question illustration
  1. $2$
  2. $3$
  3. $\pi$
  4. $4$
  5. $8$

Question 10

In 2014 all four digits differ, and the last exceeds the sum of the other three. How many years earlier did this last happen?
  1. 5
  2. 215
  3. 305
  4. 395
  5. 485

Question 11

A cuboid has side lengths $a\lt b\lt c$. Increasing one side by 5 cm increases its volume. Which choice gives the greatest increase?
  1. Increase a
  2. Increase b
  3. Increase c
  4. It depends on a,b,c
  5. All three increases are equal

Question 12

Four football teams A,B,C,D play each other once. A win gives 3 points, a loss 0, a draw 1 to each team. A finishes with 7 points, B and C with 4 each. How many points has D?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 13

Two concentric circles have radii in ratio 1:3. AC is a diameter of the larger circle; its chord BC is tangent to the smaller. If AB=12, find the larger radius.

Question illustration
  1. 13
  2. 18
  3. 21
  4. 24
  5. 26

Question 14

How many integer triples $(a,b,c)$ with $a>b>c>1$ satisfy $1/a+1/b+1/c>1$?
  1. None
  2. 1
  3. 2
  4. 3
  5. Infinitely many

Question 15

Six weeks equal n! seconds, where n!=n(n−1)…2·1. Find n.
  1. 6
  2. 7
  3. 8
  4. 10
  5. 12

Question 16

The cube vertices are numbered 1 to 8, once each, so that every face has the same sum of its four vertex numbers. The positions of 1,4,6 are shown. Find x.

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

Question 17

Soft cheese has 24% total fat, and its dry matter has 64% fat. What percentage of the cheese is water?
  1. 88%
  2. 62.5%
  3. 49%
  4. 42%
  5. 37.5%

Question 18

For f(x)=ax+b, we know f(f(f(1)))=29 and f(f(f(0)))=2. Find a.
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 19

Among ten distinct positive integers, exactly five are divisible by 5 and exactly seven by 7. If M is the largest, what is its smallest possible value?
  1. 105
  2. 77
  3. 75
  4. 63
  5. Another value

Question 20

PQRS is a rectangle. T is the midpoint of RS, and QT is perpendicular to diagonal PR. Find PQ:QR.

Question illustration
  1. $2:1$
  2. $\sqrt3:1$
  3. $3:2$
  4. $\sqrt2:1$
  5. $5:4$