IKMC 2020 - Grades 11-12 - Set 2 - Self-study

What is the sum of the last two digits of $1\cdot2\cdot3\cdot4\cdot5\cdot4\cdot3\cdot2\cdot1$?

An ant walked every day along a straight horizontal path from A to B, 5 m apart. Two obstacles, each of height 1 m, are placed on its path as shown. It still follows the same straight line in horizontal projection, climbing vertically up and down over both obstacles. How long is its new path?

Rene marked points $a$ and $b$ as accurately as possible on the number line. Which of $p,q,r,s,t$ best represents the product $ab$?

The pie chart shows how the students of my school get to school. Approximately twice as many go by bike as use public transport, and roughly the same number come by car as walk. The rest use a moped. What percentage use a moped?

The sum of five three-digit numbers is 2664, as shown on the board. What is $A+B+C+D+E$?

What is $\dfrac{1010^2+2020^2+3030^2}{2020}$?

Integers $a,b,c$ satisfy $1\le a\le b\le c$ and $abc=1000000$. What is the largest possible value of $b$?

If $D$ dogs weigh $K$ kilograms and $E$ elephants weigh the same as $M$ dogs, how many kilograms does one elephant weigh?

Each of two dice has two red faces, two blue faces and two white faces. If both fair dice are rolled together, what is the probability they show the same colour?

Which expression is not divisible by 3 for any integer $n$?

A blue rectangle and a red rectangle overlap in four different ways as shown. Let $B$ be the area of the blue part outside the overlap, and $R$ the area of the red part outside the overlap. Which statement about $B-R$ is true?

Five coins lie heads up. At each step you must turn over exactly three coins. What is the least number of steps needed to make all five coins tails up?

Four identical cuboid boxes are glued together in the shape shown. One litre of paint is needed to paint the entire outside of one box. How many litres are needed to paint the outside of the glued construction?

For integers $a,b,c$, which value certainly cannot equal $(a-b)^2+(b-c)^2+(c-a)^2$?

The first two digits of a 100-digit positive integer are 2 and 9. How many digits does its square have?

Matjaz places 15 numbers on a wheel. Only the top number, 10, is visible. Any seven consecutive positions always have the same sum, as in the shaded example. How many of 75, 216, 365 and 2020 can be the total of all 15 numbers?

A large square touches two other squares as shown. The numbers 9 and 1 in the small squares are their areas. What is the area of the large square?

The sequence has $f_1=1$, $f_2=3$, and $f_{n+2}=f_n+f_{n+1}$ for $n\ge1$. How many of its first 2020 terms are even?

A circle touches two sides of a rectangle and passes through one vertex, as shown. The distances along the bottom side from the tangency point to the two bottom vertices are 5 and 4. What is the rectangle’s area?

Three cuboids form a larger cuboid as shown. One width is 6, and the indicated face areas are 14, 21, 16 and 30. What is the area of the face marked with a question mark?