IKMC 2020 - Grades 9-10 - Self-study

The shape is made from ten unit squares of side 1 cm joined edge to edge. What is its perimeter?

Put the answers to these calculations in increasing order. Which is in the middle?

Who is the mother of the daughter of Anne’s mother’s mother?

When Cosmo buttons his shirt correctly, the horizontal stripes form seven closed rings around his waist, as on the left. This morning he buttoned it incorrectly, as on the right. How many closed rings were then around his waist?

Each letter represents a digit, forming the two-digit numbers shown. The two numbers on the left sum to 79. What is the sum of the four numbers on the right?

Four consecutive integers have sum 2. What is the smallest?

The years 2020 and 1717 each consist of a two-digit number repeated twice. How many years after 2020 is the next year with this property?

Mary has ten pieces of paper, some squares and the rest triangles. She cuts three squares along a diagonal. Her thirteen resulting pieces have 42 vertices in total. How many triangles did she have before cutting?

Martin cuts a wooden pole into six pieces to make a kite. Two pieces,120 cm and 80 cm, are its diagonals. The other four join the midpoints of adjacent sides as shown. How long was the original pole?

Four points are marked on a unit-square grid. What is the smallest area of a triangle whose vertices are three of these points?

Helen visits her Grandma for 18 consecutive days. Grandma reads stories on Tuesdays, Saturdays and Sundays. On which day should Helen start to have as many story days as possible?

Integers a, b, c, d satisfy ab=2 cd. Which number cannot equal abcd?

The shortest route from Atown to Cetown passes through Betown. Walking along it, the first signpost is the left one and later the right one. What distance was on the broken Betown sign?

An isosceles triangle has a side 20 cm. Of the other two sides, one is 2/5 of the other. Which listed value is its perimeter?

Tom writes a number in each of the nine cells. The three numbers on every diameter sum to 13, and the eight numbers around the circumference sum to 40. What is the central number?

Masha inserts a multiplication sign between the second and third digits of 2020 and obtains the square 20 × 20. How many numbers from 2010 to 2099, including 2020, have this property?

Two squares of different sizes lie inside an equilateral triangle. One side of the smaller square lies on a triangle side as shown. Given the 70 degree angle, find the angle marked “? ”.

Luca starts a 520 km trip with 14 litres of fuel. His car uses 1 litre per 10 km. After 55 km a sign lists five stations 35,45,55,75 and 95 km ahead. The tank holds 40 litres. He wants to fill the tank at just one station and finish the trip. Which listed station should he use?

If 17 x+51 y=102, what is 9 x+27 y?

A square stained-glass window of area 81 dm² consists of six equal-area triangles. A fly sits where all six meet. How far is the fly from the bottom edge?