2017 Auckland Mathematical Olympiad

Part of Auckland Mathematical Olympiad

Subcontests

(5)

2

average mean, integers 1 to n related (2017 Auckland MO J4)

The positive integers from

1

n

inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining

n - 1

22

n

is odd, determine

n

and the number that was erased. Explain your reasoning.

11 empty boxes and a pile of stones (2017 Auckland MO S4)

11

empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes

10

stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear

21

stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

2

2 player game with divisors of naturals (2017 Auckland MO J2)

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than

p

and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for

p = 10

and describe this strategy.

(1 - x_1)(1 - x_2)(1 - x_3) >= 1/2 if x_1+x_2+x_3<=1/2 (2017 Auckland MO S2)

The sum of the three nonnegative real numbers

x_1, x_2, x_3

is not greater than

\frac12

(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12

2

a ladder rests against a vertical wall (2017 Auckland MO J1)

6

meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?

more parents than children in the building? (2017 Auckland MO S1)

In an apartment block there live only couples of parents with children. It is known that every couple has at least one child, that every child has exactly two parents, that every little boy in this building has a sister, and that among the children there are more boys than girls. You may also assume that there are no grandparents living in the building. Is it possible that there are more parents than children in the building? Explain your reasoning.

2

rectangles by lines // sides of rectangle (2017 Auckland MO J5)

ABCD

is given. On the side

AB

, n different points are chosen strictly between

A

B

m

different points are chosen on the side

AD

A

D

. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way?
An example of a particular rectangle

ABCD

is shown with a shaded one rectangle that may be formed in this way. https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png

<C wanted, AB = CH, orthocenter (2017 Auckland MO S5)

The altitudes of triangle

ABC

intersect at a point

H

\angle ACB

if it is known that

AB = CH

2

3 equal circles,each pass through centers of other 2 (2017 Auckland MO J3)

Three equal circles of radius

r

each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?

N = 11...11 divisible by 13 if divisible by 7 (2017 Auckland MO S3)

The positive integer

N = 11...11

, whose decimal representation contains only ones, is divisible by

7

. Prove that this positive integer is also divisible by

13