MathDB

Round 1
p1. Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city?
p2. Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea?
p3. Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off?
p4. Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law.
p5. Alice and Bob split the numbers from

1

12

into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as

a_1 < a_2 < a_3 < a_4 < a_5 < a_6

and Bob lists the numbers in the second pile in decreasing order

b_1 > b_1 > b_3 > b_4 > b_5 > b_6

. Show that no matter how they split the numbers,

|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.

Round 2
p6. The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written?
p7. For a long space journey, two astronauts with compatible personalities are to be selected from

24

candidates. To find a good fit, each candidate was asked

24

questions that required a simple yes or no answer. Two astronauts are compatible if exactly

12

of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these

24

astronauts was compatible! Show that there were exactly

12

astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Round 1
p1. Alice and Bob played

25

games of rock-paper-scissors. Alice played rock

12

times, scissors

6

times, and paper

7

times. Bob played rock

13

times, scissors

9

times, and paper

3

times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.)
p2. On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were

30

eruptions. How many volcanoes could erupt this year?
p3. A tangle is a sequence of digits constructed by picking a number

N\ge 0

and writing the integers from

0

N

in some order, with no spaces. For example,

010123459876

is a tangle with

N = 10

. A palindromic sequence reads the same forward or backward, such as

878

6226

. The shortest palindromic tangle is

0

. How long is the second-shortest palindromic tangle?
p4. Balls numbered

1

N

have been randomly arranged in a long input tube that feeds into the upper left square of an

8 \times 8

board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from

1

N

in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number

N

for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png
p5. A

2018 \times 2018

board is covered by non-overlapping

2 \times 1

dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png
Round 2
p6. Seventeen teams participated in a soccer tournament where a win is worth

1

point, a tie is worth

0

points, and a loss is worth

-1

point. Each team played each other team exactly once. At least

\frac34

of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament.
p7. The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are

2018

influential secret societies. Show that it is possible to form a council of at most

11

people such that each influential secret society has at least one member on the council.

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

2018 Math Hour Olympiad

Subcontests

2018 Math Hour Olympiad - University of Washington - Grades 8-10

2018 Math Hour Olympiad - University of Washington - Grades 6-7