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Math Hour Olympiad
2022 Math Hour Olympiad
2022 Math Hour Olympiad
Part of
Math Hour Olympiad
Subcontests
(2)
8-10
1
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2022 Math Hour Olympiad - University of Washington - Grades 8-10
Round 1 p1. Alex is writing a sequence of
A
A
A
’s and
B
B
B
’s on a chalkboard. Any
20
20
20
consecutive letters must have an equal number of
A
A
A
’s and
B
B
B
’s, but any 22 consecutive letters must have a different number of
A
A
A
’s and
B
B
B
’s. What is the length of the longest sequence Alex can write?. p2. A positive number is placed on each of the
10
10
10
circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all
10
10
10
numbers are different? https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png p3. Pablo and Nina take turns entering integers into the cells of a
3
×
3
3 \times 3
3
×
3
table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to
0
0
0
. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? p4. All possible simplified fractions greater than
0
0
0
and less than
1
1
1
with denominators less than or equal to
100
100
100
are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “
+
+
+
” or “
−
-
−
” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.pngp5. 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/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png Round 2p6. Prove that among any
3
2022
3^{2022}
3
2022
integers, it is possible to find exactly
3
2021
3^{2021}
3
2021
of them whose sum is divisible by
3
2021
3^{2021}
3
2021
. p7. Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is
(
5
,
7
,
10
)
(5, 7, 10)
(
5
,
7
,
10
)
and you zap
5
5
5
and
10
10
10
, whose average is
7.5
7.5
7.5
, the new list is
(
−
2.5
,
7
,
2.5
)
(-2.5, 7, 2.5)
(
−
2.5
,
7
,
2.5
)
. Is it possible to start with the list
(
3
,
1
,
4
)
(3, 1, 4)
(
3
,
1
,
4
)
and, through some sequence of zaps, end with a list in which the sum of the three numbers is
0
0
0
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
6-7
1
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2022 Math Hour Olympiad - University of Washington - Grades 6-7
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
A
A
’s and
B
B
B
’s on a chalkboard. Any
20
20
20
consecutive letters must have an equal number of
A
A
A
’s and
B
B
B
’s, but any 22 consecutive letters must have a different number of
A
A
A
’s and
B
B
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.pngp4. 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
64
64
moves on an
8
×
8
8\times 8
8
×
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
1, 2,..., 100
1
,
2
,
...
,
100
in a
10
×
10
10 \times 10
10
×
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
30
30
and no more than
71
71
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
101
101
cards numbered
1
1
1
through
101
101
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
101
101
was played against card
2
2
2
on the
10
10
10
th move. Prove that this game will never end. https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.pngp7. The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a
10
×
10
10 \times 10
10
×
10
grid, crushing any teacups on roofs of buildings within a
5
×
1
5 \times 1
5
×
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
4
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.