IKMC 2023 - Grades 7-8 - Self-study

The diagram shows a set of horizontal and vertical lines with one part removed. Which of the following could be the missing part?

Which of the shapes below cannot be divided into two trapezia by a single straight line? A: triangle; B: rectangle; C: trapezium; D: regular hexagon; E: square.

A grey circle with two holes in it is placed on top of a clock-face, as shown. The grey circle is turned around its centre such that an 8 appears in one of the holes. Which two numbers could be seen in the other hole?

Werner wants to write a number at each vertex and on each edge of the rhombus shown. He wants the sum of the numbers at the two vertices at the ends of each edge to be equal to the number written on the edge. What number will he write instead of the question mark?

Kristina has a piece of transparent paper with some lines marked on it. She folds it along the dashed line. What can she now see?

A tiler wants to tile a floor of dimensions $4\,\mathrm m\times6\,\mathrm m$ using identical tiles. No overlaps or gaps are allowed. Which of the following tiles could not be used?

John has 150 coins. When he throws them on the table, 40% of them show heads and 60% of them show tails. How many coins showing tails does he need to turn over to have the same number show heads as show tails?

The diagram shows the initial position, the direction of travel and how far four bumper cars move in five seconds. Which two of the cars will collide?

Anna has five circular discs, each of a different size. She decides to build a tower using three of her discs so that each disc in her tower is smaller than the disc below it. How many different towers could Anna construct?

Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3, 4 and 8, as shown. What number will she write in the shaded box?

Theodorika wrote down three consecutive whole numbers in order, but instead of digits she used symbols, as shown. What would she write next?

The diagram shows five equal semicircles and the lengths of some line segments. What is the radius of the semicircles?

Some edges of a cube are to be coloured red so that every face of the cube has at least one red edge. What is the smallest possible number of edges that could be coloured red?

Matchsticks can be used to write digits, as shown in the diagram. How many different positive integers can be written using exactly six matchsticks in this way?

The edges of a square are 1 cm long. How many points on the plane are exactly 1 cm away from two of the vertices of this square?

Triangle ABC is isosceles with $\angle ABC=40^\circ$. The two marked angles, $\angle EAB$ and $\angle DCA$, are equal. What is the size of angle CFE?

Tom, John and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring score the same number of points. Tom scored 46 points and John scored 34 points, as shown. How many points did Lily score?

The diagram shows a rectangle made from three grey squares, each of area $25\,\mathrm{cm}^2$, inside a larger white rectangle. Two of the vertices of the grey rectangle touch the midpoints of the shorter sides of the white rectangle and the other two vertices of the grey rectangle touch the other two sides of the white rectangle. What is the area, in $\mathrm{cm}^2$, of the white rectangle?

Angelo drew two lines meeting at a right angle. What is the smallest number of extra lines he could draw inside his right angle, as shown, so that for any of the values $10^\circ,20^\circ,30^\circ,40^\circ,50^\circ,60^\circ,70^\circ,80^\circ$, a pair of lines can be chosen with the angle between them equal to that value?

The sum of 2023 consecutive integers is 2023. What is the sum of digits of the largest of these integers?