MathDB

Round 1
p1. Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”?
p2. Alex is writing a sequence of

A

’s and

B

’s on a chalkboard. Any

20

consecutive letters must have an equal number of

A

’s and

B

’s, but any 22 consecutive letters must have a different number of

A

’s and

B

’s. What is the length of the longest sequence Alex can write?.
p3. A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png
p4. A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of

64

moves on an

8\times 8

chessboard so that it visits each square exactly once and returns to its starting position? https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png
p5. Ann places the integers

1, 2,..., 100

in a

10 \times 10

grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least

30

and no more than

71

. Can he find a strategy to do this, no matter how Ann originally arranged the numbers?
Round 2
p6. Evelyn and Odette are playing a game with a deck of

101

cards numbered

1

through

101

. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card

101

was played against card

2

on the

10

th move. Prove that this game will never end. https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png
p7. The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a

10 \times 10

grid, crushing any teacups on roofs of buildings within a

5 \times 1

length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear

4

cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement