IKMC 2013 - Grades 7-8 - Self-study

Question 1

In the picture, the big triangle is equilateral and has area 9. The lines are parallel to the sides and divide the sides into three equal parts. What is the area of the shaded part?

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Question 2

It is true that $\frac{1111}{101}=11$. What is the value of $\frac{3333}{101}+\frac{6666}{303}$?

Question 3

The masses of salt and fresh water in sea water in Protaras are in the ratio 7:193. How many kilograms of salt are there in 1000 kg of sea water?

Question 4

Ann has the square sheet of paper shown on the left. By cutting along the lines of the square, she cuts out copies of the shape shown on the right. What is the smallest possible number of cells remaining?

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Question 5

Roo wants to tell Kanga a number with the product of its digits equal to 24. What is the sum of the digits of the smallest number that Roo could tell Kanga?

Question 6

A bag contains balls of five different colours: two red, three blue, ten white, four green and three black. Balls are taken without looking and not returned. What is the smallest number to take to be sure that two balls of the same colour have been taken?

Question 7

Alex lights a candle every ten minutes. Each candle burns for 40 minutes and then goes out. How many candles are alight 55 minutes after Alex lit the first candle?

Question 8

The average number of children in five families cannot be

Question 9

Mark and Liza stand on opposite sides of a circular fountain. They start running clockwise round it. Mark’s speed is $\frac98$ of Liza’s speed. How many circuits has Liza completed when Mark first catches her?

Question 10

Positive integers $x,y,z$ satisfy $xy=14$, $yz=10$ and $zx=35$. What is $x+y+z$?

Question 11

Carina and a friend play battleships on a 5×5 board. Carina has placed two ships as shown. She still has to place a 3×1 ship covering exactly three cells. No two ships can have a point in common. How many positions are there for her 3×1 ship?

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Question 12

In the diagram, $\alpha=55^\circ$, $\beta=40^\circ$ and $\gamma=35^\circ$. What is $\delta$?

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$100^\circ$
$105^\circ$
$120^\circ$
$125^\circ$
$130^\circ$

Question 13

The perimeter of a trapezium is 5 and its side lengths are integers. What are its smallest two angles?

$30^\circ,30^\circ$
$60^\circ,60^\circ$
$45^\circ,45^\circ$
$30^\circ,60^\circ$
$45^\circ,90^\circ$

Question 14

Which of these nets cannot be folded to form a cube?

Question 15

Vasya wrote several consecutive integers. Which could not be the percentage of odd numbers among them?

Question 16

The edges of rectangle ABCD are parallel to the coordinate axes. It lies below the x-axis and to the right of the y-axis, as shown. All vertex coordinates are integers. For each vertex, calculate y-coordinate ÷ x-coordinate. Which gives the least value?

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A
B
C
D
It depends on the rectangle.

Question 17

All four-digit positive integers using the same four digits as 2013 are written in increasing order. What is the greatest difference between neighbouring numbers?

Question 18

In the shown 6×8 grid, 24 cells are not intersected by either diagonal. When both diagonals of a 6×10 grid are drawn, how many cells are not intersected by either diagonal?

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Question 19

Andy, Betty, Cathie, Dannie and Eddy were born on 20/02/2001, 12/03/2000, 20/03/2001, 12/04/2000 and 23/04/2001 (day/month/year). Andy and Eddy were born in the same month, as were Betty and Cathie. Andy and Cathie have the same day number in different months, as do Dannie and Eddy. Who is the youngest?

Andy
Betty
Cathie
Dannie
Eddy

Question 20

John has made a building of cubes on a 4×4 grid. The diagram gives the number of cubes on each cell. What does John see from the back?

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