IKMC 2018 - Grades 11-12 - Set 2 - Self-study

Question 1

In how many ways can 1001 be written as the sum of two prime numbers?

None
One
Two
Three
More than three

Question 2

Thor has seven stones. Every hammer strike breaks one stone into exactly five smaller stones. After several strikes, which of these could be the number of stones?

Question 3

Positive integers a, b satisfy $2^a-2^b=992$. Find a+b.

Question 4

How many squares with integer side length have area $\dfrac{n(3n+1)}3$, where n is a positive integer?

None
One
Two
Three
Infinitely many

Question 5

How many real solutions satisfy $2^x+8^x=2\cdot5^x$?

0
1
2
3
Infinitely many

Question 6

Some aliens are green and the others purple. Green aliens live only on Mars. Which statement logically follows?

All aliens live on Mars
Only green aliens live on Mars
Some purple aliens live on Venus
All purple aliens live on Venus
No green aliens live on Venus

Question 7

Four identical rhombuses and two squares form the regular octagon shown. Find the larger angle of each rhombus.

Question illustration

$135^\circ$
$140^\circ$
$144^\circ$
$145^\circ$
$150^\circ$

Question 8

A box contains 65 balls:8 white and 57 black. In one move at most 5 balls may be removed, and none may be returned. What is the fewest moves guaranteeing at least one white ball?

Question 9

The three adjacent face areas of a rectangular brick are A, B, C as shown. What is its volume?

Question illustration

$ABC$
$\sqrt{ABC}$
$\sqrt{AB+BC+CA}$
$\sqrt[3]{ABC}$
$2(A+B+C)$

Question 10

What is the sum of the eight marked angles in the figure?

Question illustration

$360^\circ$
$450^\circ$
$540^\circ$
$720^\circ$
$900^\circ$

Question 11

Two cubes have volumes V and W and overlap. The part of the V-cube outside the overlap is 90% of V; the part of the W-cube outside it is 85% of W. Which relation holds?

Question illustration

$V=\frac23W$
$V=\frac32W$
$V=\frac{85}{90}W$
$V=\frac{90}{85}W$
$V=W$

Question 12

A vase fills to the top with water at a constant volume rate. The graph gives water height h against time t. Which pictured shape could be the vase?

Question illustration

Question 13

How many ordered pairs of real numbers (a, b) satisfy $a+b=ab=a/b$?

1
2
3
Infinitely many
None

Question 14

An octahedron is inscribed in a unit cube, with its six vertices at the centres of the cube faces. Find its volume.

Question illustration

$1/3$
$1/4$
$1/5$
$1/6$
$1/8$

Question 15

Triangle vertices are A(p, q), B(r, s), C(t, u). The side midpoints are M(−2,1), N(2,−1), P(3,2) as shown. Find p+q+r+s+t+u.

Question illustration

2
$5/2$
3
5
None of these

Five predictions were made for Real Madrid–Manchester United:

1. The game will not be a draw.
2. Real Madrid will score.
3. Real Madrid will win.
4. Real Madrid will not lose.
5. Exactly three goals will be scored.

Exactly three predictions were true. Which final score is possible, with Real Madrid first?