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IKMC 2013 - Grades 11-12 - Paper 01 - Self-study

Question 1

Which number is largest?
  1. $2013$
  2. $2^{0+13}$
  3. $20^{13}$
  4. $201^3$
  5. $20\cdot13$

Question 2

A regular octagon has side length 10. Find the radius of the circle inscribed in the smallest octagon formed by the diagonals shown.

Question illustration
  1. 10
  2. 7.5
  3. 5
  4. 2.5
  5. 2

Question 3

A prism has 2013 faces in total. How many edges does it have?
  1. 2011
  2. 2013
  3. 4022
  4. 4024
  5. 6033

Question 4

What is the cube root of $3^{3^3}$?
  1. $3^3$
  2. $3^{3^3-1}$
  3. $3^{2^3}$
  4. $3^{3^2}$
  5. $(\sqrt3)^3$

Question 5

The year 2013 uses the four consecutive digits 0, 1, 2, 3. How many years have passed since the previous year whose four digits were consecutive?
  1. 467
  2. 527
  3. 581
  4. 693
  5. 990

Question 6

A linear function f satisfies f(2013)−f(2001)=100. Find f(2031)−f(2013).
  1. 75
  2. 100
  3. 120
  4. 150
  5. 180

Question 7

For $2<x<3$, how many of these statements are true? $4<x^2<9$; $4<2x<9$; $6<3x<9$; $0<x^2-2x<3$.
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 8

Six superheroes capture 20 villains. The first captures 1, the second 2, the third 3. The fourth captures more than each of the other five. What is the smallest possible number captured by the fourth?
  1. 7
  2. 6
  3. 5
  4. 4
  5. 3

Question 9

Inside the cube is an opaque solid pyramid ABCDS with base ABCD. Its apex S is the midpoint of the indicated cube edge. Viewed from above, below, behind, in front, right and left, which drawing never occurs?

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Question 10

When a solid melts, its volume increases by 1/12 of its original volume. By what fraction does its volume decrease when it solidifies again?
  1. $1/10$
  2. $1/11$
  3. $1/12$
  4. $1/13$
  5. $1/14$

Question 11

Radu glues identical regular-pentagon pieces edge to edge to complete a circular ring, continuing the pattern shown. How many pieces form the ring?

Question illustration
  1. 8
  2. 9
  3. 10
  4. 12
  5. 15

Question 12

For how many positive integers n are both n/3 and 3n three-digit integers?
  1. 12
  2. 33
  3. 34
  4. 100
  5. 300

Question 13

A circular carpet lies on a floor of square tiles. Every tile having more than one point in common with the carpet is grey. Which outcome is impossible?
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Question 14

For a function on the integers, consider: “For every even x, f(x) is even.” What is its negation?
  1. For every even x, f(x) is odd.
  2. For every odd x, f(x) is even.
  3. For every odd x, f(x) is odd.
  4. There exists an even x with f(x) odd.
  5. There exists an odd x with f(x) odd.

Question 15

For $W(x)=(a-x)(b-x)^2$, where $a<b$, which graph is correct?
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Question 16

One side of a rectangle has length 5. It can be cut into a square and a rectangle, one of which has area 4. How many different rectangles satisfy this?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 17

The graph of f: R→R consists of two rays and a segment as shown. How many solutions does f(f(f(x)))=0 have?

Question illustration
  1. 4
  2. 3
  3. 2
  4. 1
  5. 0

Question 18

In triangle ABC, M and N lie on AB as shown, with AN=AC and BM=BC. If angle MCN is 43°, find angle ACB.

Question illustration
  1. 86°
  2. 89°
  3. 90°
  4. 92°
  5. 94°

Question 19

How many ordered pairs of positive integers $(x,y)$ satisfy $x^2y^3=6^{12}$?
  1. 6
  2. 8
  3. 10
  4. 12
  5. Another number

Question 20

A box contains the 900 distinct cards numbered 100 to 999. François draws cards and finds each digit sum. At least how many must he draw to guarantee three with equal digit sums?
  1. 51
  2. 52
  3. 53
  4. 54
  5. 55