IKMC 2018 - Grades 9-10 - Self-study

Every child in my family has at least two brothers and at least one sister. What is the smallest possible number of children?

How many of these four numbers are multiples of 10: a) $2^{100}+2^{102}$; b) $3^{100}+3^{102}$; c) $5^{100}+5^{101}$; d) $9^{100}+9^{101}$?

Two sides of a triangle have lengths 5 and 2. The third side is an odd integer. Find its length.

From the top of a sleeping cat on the floor to the top of a cat sitting on a table is 150 cm. From the top of a cat sitting on the floor to the top of a cat sleeping on the table is 110 cm. How high is the table?

Five consecutive integers have sum $10^{2018}$. What is the middle integer?

Three congruent regular hexagons have shaded areas X,Y,Z as shown. Which statement is true?

Mary has 42 apples, 60 apricots and 90 cherries. She uses all the fruit to make identical piles, one for each friend. What is the greatest possible number of piles?

In the correct addition shown, letters P,Q,R,S replace digits. Find P+Q+R+S.

What is the sum of 25% of 2018 and 2018% of 25?

Following the arrows in the diagram, how many different routes lead from A to B?

Two buildings on a street are 250 m apart. The first houses 100 students, the second 150. Where should a bus stop be placed to minimise the total walking distance from the stop to their buildings for all residents?

A row contains 105 numbers: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,…, with each n appearing exactly n times. How many entries are divisible by 3?

Eight congruent semicircles lie inside a square of side 4 as shown. What is the area of the non-shaded part, including the white corner lenses?

Forty trains each travel between two of towns M,N,O,P,Q. Exactly 10 trains arrive at or depart from each of M,N,O,P. How many trains arrive at or depart from Q?

A humanities university offers languages, history and philosophy. Of language students, 35% study English. Of all students, 13% study a language other than English. Nobody studies more than one language. What percentage of all students study languages?

Peter has no money for a book. His father and two brothers pay for it. Father gives half the amount given by both brothers together. The older brother gives one third of what the others give together. The younger brother gives 10 euros. What is the book’s price?

How many three-digit numbers become one ninth of themselves when their middle digit is deleted?

How many copies of $2018^2$ occur inside the square root in the correct equality $\sqrt{2018^2+2018^2+\cdots+2018^2}=2018^{10}$?

How many decimal digits does $\frac19\times10^{2018}\times(10^{2018}-1)$ have?

Vertices of a regular 2018-gon are numbered 1 through 2018 in order. Draw diagonals 18–1018 and 1018–2000. How many vertices do the three resulting polygons have?