IKMC 2007 - Grades 11-12 - Self-study

Question 1

Awais is building a race track. He notices that the order of the cars at the end is not the same as at the beginning. Which element should replace element $X$ at the beginning so that the cars finish in their original order?

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Question 2

Three boys have 30 balls altogether. Arshad gives 5 to Babar, Babar gives 4 to Ejaz, and Ejaz gives 2 to Arshad. They then have equal numbers of balls. How many balls did Ejaz have at the beginning?

Question 3

The shaded area is $\sqrt3$. What is the area of triangle $ABC$?

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$2\sqrt3$
2
5
4
$4\sqrt3$

Question 4

Calculate $\dfrac{\sin1^\circ}{\cos89^\circ}$.

0
$\tan1^\circ$
$\cot1^\circ$
$\frac1{89}$
1

Question 5

The billiard ball meets the cushion at $45^\circ$, as shown. Which pocket will it fall into?

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A
B
C
D
None of the pockets

Question 6

Some historians claim that ancient Egyptians used a string with two knots to construct a right angle. The string is 12 m long, and knot $X$ is 3 m from one end. How far from the other end should the second knot be placed to obtain a right angle at $X$?

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3
4
5
6
Another answer

Question 7

A university entrance test requires at least 80% correct answers. Raza has worked on 15 questions: he did not know the answers to 5, but is sure that the other 10 are correct. If he answers all the remaining questions correctly, he will pass with exactly 80%. How many questions are in the test?

Question 8

Which objects can be obtained by rotating the given object in space?

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W and Y
X and Z
Only Y
None of these
W, X and Y

Question 9

$AE$ is divided into four equal parts. Semicircles with diameters $AE$, $AD$, and $DE$ form the two paths from $A$ to $E$ shown. Find the ratio of the upper path’s length to the lower path’s length.

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1 : 2
2 : 3
2 : 1
3 : 2
1 : 1

Question 10

A spider’s web has the lengths marked in the diagram. If $x$ is an integer, find $x$.

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Question 11

A square $ABCD$ has side length 1. Draw all squares that share at least two vertices with $ABCD$. What is the area of the region covered by at least one of these squares?

Question 12

Angle $\beta$ is 25% less than angle $\gamma$ and 50% greater than angle $\alpha$. Angle $\gamma$ is:

25% greater than α
50% greater than α
75% greater than α
100% greater than α
125% greater than α

Question 13

Given $2^{x+1}+2^x=3^{y+2}-3^y$, where $x$ and $y$ are integers, find $x$.

0
3
−1
1
2

Question 14

Find $\cos1^\circ+\cos2^\circ+\cdots+\cos358^\circ+\cos359^\circ$.

1
$\pi$
0
10
−1

Question 15

Two semicircles are drawn as shown. Chord $CD$ of the larger semicircle has length 4, is parallel to its diameter $AB$, and is tangent to the smaller semicircle. What is the shaded area?

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$\pi$
$1.5\pi$
$2\pi$
$3\pi$
Not enough information

Question 16

The sum of five consecutive integers equals the sum of the next three consecutive integers. What is the greatest of these eight numbers?

4
8
9
11
Something else

Question 17

Akhlaq was born on his mother’s 20th birthday, so they share birthdays. How many times will Akhlaq’s age be a divisor of his mother’s age if they both live long lives?

Question 18

An island is inhabited by knights, who always tell the truth, and liars, who always lie. Islander A says that at least one of A and B is a liar. Which statement is true?

A cannot make that statement.
Both are liars.
Both are knights.
A is a liar and B is a knight.
B is a liar and A is a knight.

Question 19

A circle has radius 5 and centre at the origin of a Cartesian coordinate system. How many points on it have integer coordinates?

Question 20

Which graph represents $y=\sqrt{\left|(1+x)(1-|x|)\right|}$?