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Math Hour Olympiad
2023 Math Hour Olympiad
2023 Math Hour Olympiad
Part of
Math Hour Olympiad
Subcontests
(2)
6-7
1
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2023 Math Hour Olympiad - University of Washington - Grades 6-7
Round 1p1. Ash is running around town catching Pokémon. Each day, he may add
3
,
4
3, 4
3
,
4
, or
5
5
5
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
100
100
100
Pokémon? p2. Jack and Jill have ten buckets. One bucket can hold up to
1
1
1
gallon of water, another can hold up to
2
2
2
gallons, and so on, with the largest able to hold up to
10
10
10
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
2023
2023
2023
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
101
101
101
cards numbered
1
,
2
,
.
.
.
,
101
1, 2, ..., 101
1
,
2
,
...
,
101
. She starts with
50
50
50
random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all
50
50
50
cards in her hand with her choice of
50
50
50
of the
51
51
51
cards from the table. Show that Camila can make at most 50 exchanges and end up with cards
1
,
2
,
.
.
.
,
50
1, 2, ..., 50
1
,
2
,
...
,
50
. https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.pngp5. There are
101
101
101
pirates on a pirate ship: the captain and
100
100
100
crew. Each pirate, including the captain, starts with
1
1
1
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
100
100
100
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
16
16
16
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.
8-10
1
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2023 Math Hour Olympiad - University of Washington - Grades 8-10
Round 1p1. Alex is on a week-long mining quest. Each morning, she mines at least
1
1
1
and at most
10
10
10
diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either
22
22
22
or
25
25
25
. Show that she miscounted.p2. Hermione set out a row of
11
11
11
Bertie Bott’s Every Flavor Beans for Ron to try. There are
5
5
5
chocolateflavored beans that Ron likes and
6
6
6
beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? p3. There are
101
101
101
pirates on a pirate ship: the captain and
100
100
100
crew. Each pirate, including the captain, starts with
1
1
1
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? p4. There are
100
100
100
food trucks in a circle and
10
10
10
gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #
1
1
1
moves
1
1
1
truck left or right and eats there; gnome #
2
2
2
moves
2
2
2
trucks left or right and eats there; ... gnome #
10
10
10
moves
10
10
10
trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. p5. The town of Lumenville has
100
100
100
houses and is preparing for the math festival. The Tesla wiring company lays 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 hangs 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.pngRound 2p6. What is the largest number of zeros that could appear at the end of
1
n
+
2
n
+
3
n
+
4
n
1^n + 2^n + 3^n + 4^n
1
n
+
2
n
+
3
n
+
4
n
, where n can be any positive integer? p7. A tennis academy has
2023
2023
2023
members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all
2022
2022
2022
other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.