IKMC 2010 - Grades 11-12 - Paper 02 - Self-study

Question 1

The picture illustrates $1+3+5+7=4\times4$. What is $1+3+5+\cdots+19$?

Question illustration

$10\times10$
$11\times11$
$12\times12$
$13\times13$
$14\times14$

Question 2

The two rows shown have equal sums. The first row is 1,2,…,10,2010; the second is 11,12,…,20,?. What replaces the question mark?

Question illustration

1010
1020
1910
1990
2000

Two empty cubic containers have base areas $1\,\mathrm{dm}^2$ and $4\,\mathrm{dm}^2$. The small one is filled with water and used repeatedly to fill the large one. How many full small containers are needed?

Question 4

How many four-digit numbers contain only odd digits and are divisible by 5?

Question 5

A company director claims, “Every employee is at least 25 years old.” The claim is false. Which conclusion follows?

Every employee is exactly 25.
Every employee is older than 26.
No employee is 25 yet.
At least one employee is younger than 25.
At least one employee is exactly 26.

Question 6

Seven $3\times1$ bars lie in the $5\times5$ box as shown. Slide bars within the box, without rotating or removing them, to make room for one more identical bar. What is the minimum number of existing bars that must be moved?

Question illustration

Editorial clarification: Austria says “rearrange”. Scoring uses the primary paper’s sliding rules: no rotation or removal.

2
3
4
5
Impossible

Question 7

Triangle $ABC$ is right-angled at $C$, $M$ is the midpoint of hypotenuse $AB$, and the figure gives $\angle A=60^\circ$. Find $\angle BMC$.

Question illustration

$105^\circ$
$108^\circ$
$110^\circ$
$120^\circ$
$125^\circ$

Question 8

Which number could be the number of edges of a prism?

100
200
2008
2009
2010

Question 9

How many two-digit numbers with tens digit $x$ and units digit $y$ satisfy $(x-3)^2+(y-2)^2=0$?

1
2
6
32
None

Question 10

The square has side 2. The semicircles have their centres at the square’s vertices and pass through its centre. Each shaded circle has its centre on a side and is tangent to the adjacent semicircles. What is the total shaded area?

Question illustration

$4(3-2\sqrt2)\pi$
$\sqrt2\pi$
$\frac{\sqrt3}{4}\pi$
$\pi$
$\frac14\pi$

Question 11

The numbers $\sqrt7$, $\sqrt[3]7$, $\sqrt[6]7$ are consecutive terms, in that order, of a geometric progression. What is the next term?

$\sqrt[9]7$
$\sqrt[12]7$
$\sqrt[5]7$
$\sqrt[10]7$
1

Question 12

The chord AB of the larger of two concentric circles is tangent to the smaller circle. If AB=16, what is the area of the shaded annulus?

Question illustration

$32\pi$
$63\pi$
$64\pi$
$32\pi^2$
It depends on the radii

Question 13

Integers $x,y$ satisfy $2x=5y$. Which listed number can equal $x+y$?

2011
2010
2009
2008
2007

Question 14

The large equilateral triangle consists of 36 equal small equilateral triangles, each of area 1 cm². Find the area of triangle ABC indicated in the grid.

Question illustration

11 cm²
12 cm²
13 cm²
14 cm²
15 cm²

Question 15

A bag contains blue, green and red balls, at least one of each colour. Every possible draw of five balls contains at least two red balls and at least three balls of one colour. How many blue balls are in the bag?

1
2
3
4
Cannot be determined

Question 16

Which graph is the solution set of $(x-|x|)^2+(y-|y|)^2=4$?

Question 17

How many right-angled triangles have all three vertices among the vertices of a regular 14-gon?

Question 18

Replace each star in $1*2*3*4*5*6*7*8*9*10$ by + or multiplication, using ordinary operation precedence and no added parentheses. If $N$ is the largest obtainable value, what is its smallest prime factor?

2
3
5
7
Another prime

Question 19

The side lengths of a triangle are positive integers $13,x,y$ centimetres and $xy=105$. What is the perimeter?

Question 20

A paper strip is folded three times as shown. If $\alpha=70^\circ$, find $\beta$.

Question illustration

$140^\circ$
$130^\circ$
$120^\circ$
$110^\circ$
$100^\circ$