2014 Math Hour Olympiad

Part of Math Hour Olympiad

Subcontests

(8)

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2014 Math Hour Olympiad - University of Washington - Grades 5-7

Round 1
p1. Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved

3

times and was waved at twice. Bobby waved

4

times and was waved at

3

times. Cindy waved

5

times. How many times was she waved at?
p2. Sherlock and Mycroft are playing Battleship on a

4\times 4

grid. Mycroft hides a single

3\times 1

cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
p3. Thirty girls –

13

of them in red dresses and

17

in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”?
p4. Herman and Alex play a game on a

5\times 5

board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png
p5. Is it possible to find

2014

distinct positive integers whose sum is divisible by each of them?
Round 2
p6. Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses?
p7. Three warring states control the corner provinces of the island whose map is shown below. https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png As a result of war, each of the remaining

18

provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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UW Math Hour Olympiad 2014 Grades 8-10 #7

a

\lfloor a \rfloor

a

rounded down to the nearest integer. For example,

\lfloor \pi \rfloor =

3

. Show that the sequence

\lfloor \frac{2^{1}}{17} \rfloor

\lfloor \frac{2^{2}}{17} \rfloor

\lfloor \frac{2^{3}}{17} \rfloor

\dots

contains infinitely many odd numbers.

1

UW Math Hour Olympiad 2014 Grades 8-10 #6

Homer goes on the

100

100

-Donut Diet Plan specifies how many of

100

total donuts Homer will eat each day. The diet requires that the number of donuts he eats does not increase from one day to the next. For example, one

5

-day Donut Diet Plan is

40

25

25

8

2

.
Are there more

100

-Donut Diet Plans with an odd number of days or plans where Homer eats an odd number of donuts on the first day?

1

UW Math Hour Olympiad 2014 Grades 8-10 #5

An infinite number of lilypads grow in a line, numbered

\dots

-2

-1

0

1

2

\dots

Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad

0

96

. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad

0

96

, regardless of her frog's decisions?

1

UW Math Hour Olympiad 2014 Grades 8-10 #4

Hermione and Ron play a game that starts with

129

hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven.
A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses?

1

UW Math Hour Olympiad 2014 Grades 8-10 #3

2014

airports in the faraway land of Artinia. Each pair of airports is connected by a nonstop flight in one or both directions. Show that there is some airport from which it is possible to reach every other airport in at most two flights.

1

UW Math Hour Olympiad 2014 Grades 8-10 #2

A complete set of the Encyclopedia of Mathematics has

10

volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.

1

UW Math Hour Olympiad 2014 Grades 8-10 #1

Sherlock and Mycroft are playing Battleship on a

4\times4

grid. Mycroft hides a single

3\times1

cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?