MathDB

2

(a+1)/b , (b+1)/c , (a+1)/b are integers (2018 Auckland MO J5)

Find all possible triples of positive integers,

a, b, c

so that

\frac{a+1}{b}

,

\frac{b+1}{c}

and

\frac{c+1}{a}

are also integers.

sequence of numbers +1 and -1 of length n (2018 Auckland MO S5)

There is a sequence of numbers

+1

and

-1

of length

n

. It is known that the sum of every

10

neighbouring numbers in the sequence is

0

and that the sum of every

12

neighbouring numbers in the sequence is not zero. What is the maximal value of

n

?

4

2

vintage tram with boys and girls (2018 Auckland MO J4)

A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there originally were girls. How many boys and girls were there on board at the start?

2 player game with numbers <=2018 (2018 Auckland MO S4)

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding

2018

(to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

2

max difference of 2 numbers in list of 3 numbers (2018 Auckland MO J2)

Starting with a list of three numbers, the “Make-My-Day” procedure creates a new list by replacing each number by the sum of the other two. For example, from

\{1, 3, 8\}

“Make-My-Day” gives

\{11, 9, 4\}

and a new “MakeMy-Day” leads to

\{13, 15, 20\}

. If we begin with

\{20, 1, 8\}

, what is the maximum difference between two numbers on the list after

2018

consecutive “Make-My-Day”s?

times of digit 1 in N = 9 + 99 + 999 + ... +999...9 (2018 Auckland MO S2)

Consider a positive integer,

N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}

. How many times does the digit

1

occur in its decimal representation?

1

2

multiple of 2018, that starts with 2017 (2018 Auckland MO J1)

Find a multiple of

2018

whose decimal expansion's first four digits are

2017

.

a= +- b if a(x-a)^2 + b(x-b)^2 = 0 has only 1 solution (2018 Auckland MO S1)

For two non-zero real numbers

a, b

, the equation,

a(x-a)^2 + b(x-b)^2 = 0

has a unique solution. Prove that

a=\pm b

.

3

2

area of a pentagon wanted, 2 right angles (2018 Auckland MO J3)

Consider the pentagon below. Find its area. https://cdn.artofproblemsolving.com/attachments/7/b/02ad3852b72682513cf62a170ed4aa45c23785.png

area of a folded 12x18 rectangle (2018 Auckland MO S3)

A rectangular sheet of paper whose dimensions are

12 \times 18

is folded along a diagonal, creating the

M

-shaped region drawn in the picture (see below). Find the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png

2018 Auckland Mathematical Olympiad

Subcontests

(a+1)/b , (b+1)/c , (a+1)/b are integers (2018 Auckland MO J5)

sequence of numbers +1 and -1 of length n (2018 Auckland MO S5)

vintage tram with boys and girls (2018 Auckland MO J4)

2 player game with numbers <=2018 (2018 Auckland MO S4)

max difference of 2 numbers in list of 3 numbers (2018 Auckland MO J2)

times of digit 1 in N = 9 + 99 + 999 + ... +999...9 (2018 Auckland MO S2)

multiple of 2018, that starts with 2017 (2018 Auckland MO J1)

a= +- b if a(x-a)^2 + b(x-b)^2 = 0 has only 1 solution (2018 Auckland MO S1)

area of a pentagon wanted, 2 right angles (2018 Auckland MO J3)

area of a folded 12x18 rectangle (2018 Auckland MO S3)

2018 Auckland Mathematical Olympiad

Subcontests

(a+1)/b , (b+1)/c , (a+1)/b are integers (2018 Auckland MO J5)

sequence of numbers +1 and -1 of length n (2018 Auckland MO S5)

vintage tram with boys and girls (2018 Auckland MO J4)

2 player game with numbers &lt;=2018 (2018 Auckland MO S4)

max difference of 2 numbers in list of 3 numbers (2018 Auckland MO J2)

times of digit 1 in N = 9 + 99 + 999 + ... +999...9 (2018 Auckland MO S2)

multiple of 2018, that starts with 2017 (2018 Auckland MO J1)

a= +- b if a(x-a)^2 + b(x-b)^2 = 0 has only 1 solution (2018 Auckland MO S1)

area of a pentagon wanted, 2 right angles (2018 Auckland MO J3)

area of a folded 12x18 rectangle (2018 Auckland MO S3)

2 player game with numbers <=2018 (2018 Auckland MO S4)