IKMC 2027 - Grades 11-12 - Self-study SET 9 - Thematic Batch 3

Question 1

In the figure, OCDE is a rectangle with E on OB, C on OA, and D on the circle with centre O and radius OA. Given BE = 4 cm and AC = 2 cm, what is OA in centimetres?

1. 6 cm
2. 7 cm
3. 8 cm
4. 9 cm
5. Another result

Question 2

What is the 2016th digit after the decimal point in the decimal expansion of 3/101?

1. 0
2. 2
3. 3
4. 7
5. 9

Question 3

Fill each small polygon in the figure with one of 1, 2, 3, 4, one number per region. Every number must occur at least once, and any two regions sharing an edge must have different numbers. Two regions already contain 1. How many ways are there to fill the four empty regions?

1. 8
2. 10
3. 12
4. 18
5. 0

Question 4

Alice and Bob play a dice game. First they draw lots to choose who starts. They then take turns rolling a die until someone rolls a 3 or a 6; that player wins. What is the probability that Alice wins in at most two of her rolls?

Editorial clarification: The limit is two rolls by Alice herself; the die and the initial draw are fair.

1. 85/243
2. 65/162
3. 45/81
4. 39/81
5. 39/162

Question 5

Let $D=3^{2014}\cdot7^{2015}-2016$. What is the remainder when D is divided by 21?

1. 0
2. 2
3. 3
4. 7
5. 14

Question 6

A group has 23 students: 21 speak English, 17 speak Russian, and 13 speak French. What is the greatest possible number who speak all three languages?

Editorial note: The source solution confuses the maximum with the minimum. The printed question and choices are retained; see the correction below.

1. 0
2. 2
3. 5
4. 7
5. 13

Question 7

For every positive integer n, let $a_n$ be the remainder when $(n+1)^3$ is divided by $n^3$, and let $b_n$ be the remainder when $a_n$ is divided by 3. Find $b_1+b_2+\cdots+b_{2016}$.

1. 168
2. 672
3. 1008
4. 1344
5. 2014

Question 8

Anne joined several identical cubes to make the solid shown. Which of the following views cannot be obtained by rotating this solid?

Question 9

A real-valued function f has a parabolic graph through (0,0), (1,1) and (2,6). Which listed value of x satisfies f(x)=2016?

1. 20
2. 24
3. 28
4. 32
5. 36

Question 10

How many positive divisors does 2016 have?

1. 10
2. 25
3. 30
4. 36
5. 49

Question 11

Five posts stand upright in a river. Which drawing shows their reflections in the water?

Question 12

Two line segments divide triangle ABC into four non-overlapping regions as shown. Three regions have areas 1 cm², 3 cm² and 3 cm². What is the area of triangle ABC in square centimetres?

1. 12 cm²
2. 12,5 cm²
3. 13 cm²
4. 13,5 cm²
5. 14 cm²

Question 13

Two cubes of volumes V and W overlap. The non-overlapping part of the first is 90% of V, and that of the second is 85% of W. What is the ratio V:W?

1. 2/3
2. 3/2
3. 85/90
4. 90/85
5. Another answer

Question 14

A square and a triangle have equal areas. One side s of the triangle is twice the side of the square. What is the ratio of the altitude to s to the side of the square?

1. 1/4
2. 1/2
3. 1
4. 2
5. 4

Question 15

The function $f:[-5,5]\to\mathbb R$ has the graph shown. How many distinct solutions does $f(f(x))=0$ have?

1. 4
2. 5
3. 6
4. 7
5. 8

Question 16

Alex puts a number in each circle so that the sum at the three vertices is the same for each of the eight smallest triangles. What is the greatest possible number of distinct values used?

1. 1
2. 2
3. 3
4. 5
5. 8

Question 17

A tangent to the graph y=2/x meets the vertical and horizontal axes at B and A, respectively, as shown. What is the area of triangle OAB?

1. 2
2. 3
3. 4
4. 8
5. Another answer

Question 18

Mr Brown is 44 years old. His four children are 4, 6, 10 and 17 years old. After at least how many whole years will their combined age exceed his age?

Editorial convention: Compare ages after a whole number of years, as in the printed choices.

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

Question 19

An analogue clock currently shows exactly 9 o’clock. After at least how many minutes will the minute hand overtake the hour hand, rounding to the smallest whole number that satisfies the requirement?

1. 46
2. 47
3. 48
4. 49
5. 50

Question 20

If $\overline{ABCD}$ is a four-digit number with A≠0, which statement is certainly true about the cubic equation $Ax^3+Bx^2+Cx+D=\overline{ABCD}$?

1. There are three real roots: two negative and one positive.
2. There are three positive real roots.
3. There are three real roots: two positive and one negative.
4. There is exactly one real root, and it is positive.
5. There is exactly one real root, and it is negative.