IKMC 2021 - Grades 11-12 - Set 2 - Self-study

Question 1

A large square is partitioned into smaller squares as shown. A shaded circle is inscribed in every smaller square. What fraction of the large square is shaded?

Question illustration

$8\pi/9$
$13\pi/16$
$3/\pi$
$3/4$
$\pi/4$

Question 2

How many integers lie in the open interval $(20-\sqrt{21},20+\sqrt{21})$?

Question 3

A cube of edge 1 is cut into two identical cuboids. What is the surface area of one cuboid?

$3/2$
2
3
4
5

Question 4

Paula’s weather app shows predicted weather and maximum temperatures for seven days. Which graph represents the maximum temperatures in the same day order?

Question illustration

A school flagpole leans after a storm. Looking from the northwest, its tip appears to the right of its base. Looking from the east, the tip also appears to the right. Which pictured direction could it lean?

Question 6

A rectangle has length $x$ and width $y$, with $x>y$. Roll it into a cylinder’s curved surface in either possible orientation. What is the volume ratio of the longer cylinder to the shorter cylinder?

$y^2:x^2$
$y:x$
$1:1$
$x:y$
$x^2:y^2$

Question 7

Let $x=\pi/4$. Which is largest?

$x^4$
$x^2$
$x$
$\sqrt x$
$\sqrt[4]x$

Question 8

Find the area of the triangle with vertices $(p,q),(3p,q),(2p,3q)$, where $p,q>0$.

$pq/2$
$pq$
$2pq$
$3pq$
$4pq$

Question 9

How many three-digit numbers formed using only the digits 1, 3 and 5 are divisible by 3? Digits may be used more than once.

Question 10

The parabola shown is $y=ax^2+bx+c$, where $a,b,c$ are distinct real numbers. Which equation could describe the shown straight line?

Question illustration

$y=bx+c$
$y=cx+b$
$y=ax+b$
$y=ax+c$
$y=cx+a$

Question 11

How many three-digit positive integers become a three-digit integer 99 larger when their digits are reversed?

Question 12

Let $A=(0,1)\cup(2,3)$ and $B=(1,2)\cup(3,4)$, all intervals open. What is the set of all $a+b$ with $a\in A,b\in B$?

$(1,7)$
$(1,5)\cup(5,7)$
$(1,3)\cup(3,7)$
$(1,3)\cup(3,5)\cup(5,7)$
None of the previous choices

Question 13

What fraction of the positive divisors of 7! are odd?

$1/2$
$1/3$
$1/4$
$1/5$
$1/6$

Question 14

Arrange the integers 1 to 1000 in a row and compute the sums of every three adjacent entries. What is the greatest possible number of odd sums?

Question 15

The large triangle is divided into small triangles as shown. Each written number is the perimeter of that small triangle. Find the large triangle’s perimeter.

Question illustration

31
34
41
62
None of the previous choices

Question 16

For a positive integer $N$, let $p(N)$ be the product of its decimal digits. For example, $p(23)=2\times3=6$. What is $p(10)+p(11)+\cdots+p(99)+p(100)$?

2025
4500
5005
5050
None of these

Question 17

Every row and every column of the shown 5 × 5 square has the same sum. Every cell contains a number, but only some are shown. What is the missing number marked?

Question illustration

Question 18

A string lies as pictured, with three crossings hidden by coins. At each crossing, either of the two shown over-under arrangements is equally likely, independently of the others. What is the probability that pulling its ends tight produces a knot?

Question illustration

$1/2$
$1/4$
$1/8$
$3/4$
$3/8$

Question 19

A puppy pulls toilet paper from a roll at constant speed. Which graph best describes the roll thickness y as a function of unrolled length x?

Question illustration

Question 20

Squares PQRS, TUVR and UWXY meet edge-to-edge as shown. P, T, X are collinear. The areas of PQRS and TUVR are 36 and 16. Find the area of triangle PXV.

Question illustration

$44/3$
$46/3$
16
$53/3$
18