IKMC 2027 - Grades 11-12 - Self-study SET 7 - Thematic Batch 3

Question 1

A set of points forms the pigeon shown in the coordinate plane Oxy. If the coordinates of every point are swapped, (x; y) ↦ (y; x), which of the five pictures results?

Question 2

Inside square ABCD of side L cm, k smaller squares have diagonals on AC, with consecutive squares sharing a vertex as shown. Each small square has an inscribed circle. An ant may walk only along square edges or circle arcs. What is the shortest distance from A to C?

1. L(1 + π/2)
2. L($\sqrt{2}$ + π/4)
3. L(1 + π/4)
4. 2L
5. All four other choices are false.

Question 3

Five positive two-digit integers use each digit from 0 to 9 exactly once. Their sum is divisible by 10. How many different values can that sum have?

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

Question 4

The pictures show different portions of the graph of f : R → R, where f(x)=ax⁴+bx³+cx²+dx+e. What is (a²+b²)(c²+d²)+e?

1. 64
2. 8
3. −8
4. 12
5. 16

Question 5

A sequence is defined by $a_{1}$=1011 and $a_{n+1}$=(2$a_{n}$+1)/2 for every positive integer n. What is $a_{2021}$?

1. 2021
2. 2020
3. 2022
4. 1010
5. 1011

Question 6

The side lengths of a rectangle are the two roots of x²−11x+13=0, equation (5.1). What is the rectangle’s perimeter?

1. 11
2. 12
3. 13
4. 22
5. 26

Question 7

While practising Kangaroo problems, Marry can solve each 3-point problem in 2 minutes, each 4-point problem in 4 minutes, and each 5-point problem in 6 minutes. What is the greatest total score she can obtain in 75 minutes?

1. 100
2. 109
3. 110
4. 111
5. 112

Question 8

The numbers 1,2,7,9,10,15,19 are written on a board. Alex and Bob alternately erase one number until only one remains. The sum erased by Alex is twice the sum erased by Bob. Which number remains?

1. 7
2. 9
3. 10
4. 15
5. 19

Question 9

If a prism has 2016 vertices, how many edges does it have?

1. 4032
2. 3024
3. 2016
4. 1009
5. A prism cannot have 2016 vertices.

Question 10

Paula reads her weather app and draws a graph showing how the temperature changes over the coming days. Which picture matches the forecast shown?

Question 11

Alex lists, in increasing order, all positive integers formed using only the digits 1,2,3,4: 1,2,3,4,11,12,13,14,21,… . What is the 2016th term?

1. 123412
2. 123321
3. 133134
4. 133232
5. 144421

Question 12

Julia has a chocolate bar made of N small squares. At each step she eats N/2 squares if the current number N is even, and (N+1)/2 if it is odd. After four successive eating steps she has finished the bar. How many values can the initial N have?

Editorial interpretation: These are four actual eating steps; the bar first becomes empty on the fourth step.

1. 4
2. 8
3. 12
4. 16
5. 20

Question 13

In the diagram C,P,O are collinear; P,Q,V are collinear; and T,U,V are collinear. TV is a common tangent to the two circles, touching them at T and U. What is x+y?

1. 50
2. 60
3. 70
4. 80
5. 90

Question 14

Mary has black and white balls. Let n be the total number of balls and d the number of black balls minus the number of white balls. Which statement is true?

1. Both n and d are even
2. Both n and d are odd
3. One of n,d is even and the other odd
4. n and d have the same parity
5. All the preceding conclusions are incorrect

Question 15

The side lengths, in centimetres, of three squares are positive integers, one being the average of the other two. They overlap as shown, and the grey region has area 47 cm². What is the area of the largest square?

1. 529 cm²
2. 576 cm²
3. 625 cm²
4. 2209 cm²
5. 2304 cm²

Question 16

A 4×4 grid is called anti-magic if its entries are exactly the integers 1 through 16, and its four row sums, four column sums and two main-diagonal sums are ten consecutive positive integers. Some entries are given. Which number belongs in the starred cell?

1. 1
2. 2
3. 8
4. 15
5. 16

Question 17

A fair six-sided die is labelled 0,1,2,3,4,5. Alex rolls it four times and records the results in order as a decimal numeral. What is the probability that the resulting number is greater than 2021?

1. 1/1296
2. 425/648
3. 1/4
4. 1/2
5. Another answer

Question 18

Three boys start at the vertices of triangle ABC and walk in the straight directions indicated by the arrows. They meet at X and find that they have walked equal distances. What special point of the triangle is X?

1. Centroid
2. Orthocentre
3. Centre
4. Circumcentre
5. All four other choices are incorrect.

Question 19

Each of the ten vertices of the triangular grid is labelled 0,1 or 2. The sum at the vertices of every white triangle is divisible by 3, whereas the sum for every black triangle is not. Three labels are given. Which values can the question-mark vertex have?

1. 0
2. 1
3. 2
4. 1;2
5. 0; 1; 2

Question 20

On a circle of radius 1, points A,B,C,D,E,F are taken in clockwise order so that AB,BC,CD,DE,EF are respectively side lengths of inscribed regular polygons with 3,4,5,6,20 sides. How long is AF?

Editorial note: The original diagram is schematic. Use the stated regular-polygon side lengths rather than measuring the displayed separation of A and F.

1. 1/π
2. 1/(2π)
3. π/(3 + 4 + 5 + 6 + 20)
4. 1/[π(3 + 4 + 5 + 6 + 20)]
5. A and F coincide