IKMC 2021 - Grades 11-12 - Self-study

Question 1

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

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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

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?

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

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

How many three-digit numbers use only 1,3,5 and are divisible by 5? Digits may repeat.

Question 9

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 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?

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

Question 11

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

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

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 $a\in A,b\in B$?

$(1,7)$
$(1,5)\cup(5,7)$
$(1,3)\cup(3,7)$
$(1,3)\cup(3,5)\cup(5,7)$
$\varnothing$

Question 13

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

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.

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Question 16

An infinite sequence has the property that the average of its first n terms is n for every positive integer n. How many terms are less than 2021?

44
504
1010
1427
2018

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?

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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?

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

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?

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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.

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