2023 Math Hour Olympiad - University of Washington - Grades 6-7
Source:
August 7, 2023
algebrageometrycombinatoricsnumber theoryMath Hour Olympiad
Problem Statement
Round 1p1. Ash is running around town catching Pokémon. Each day, he may add , or Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly Pokémon?
p2. Jack and Jill have ten buckets. One bucket can hold up to gallon of water, another can hold up to gallons, and so on, with the largest able to hold up to gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water?
https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.pngp3. There are knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there?
p4. Camila has a deck of cards numbered . She starts with random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all cards in her hand with her choice of of the cards from the table. Show that Camila can make at most 50 exchanges and end up with cards .
https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.pngp5. There are pirates on a pirate ship: the captain and crew. Each pirate, including the captain, starts with gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate?
Round 2p6. The town of Lumenville has houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used.
https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.pngp7. You are given a sequence of digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number?
https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.pngPS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.