MathDB

8

Part of 2023 MOAA

Problems(4)

2023 MOAA Speed P8

Source:

10/13/2023
In the coordinate plane, Yifan the Yak starts at (0,0)(0,0) and makes 1111 moves. In a move, Yifan can either do nothing or move from an arbitrary point (i,j)(i,j) to (i+1,j)(i+1,j), (i,j+1)(i,j+1) or (i+1,j+1)(i+1,j+1). How many points (x,y)(x,y) with integer coordinates exist such that the number of ways Yifan can end on (x,y)(x,y) is odd?
Proposed by Yifan Kang
MOAA 2023
2023 MOAA Accuracy P8

Source:

10/13/2023
Harry wants to label the points of a regular octagon with numbers 1,2,…,81,2,\ldots ,8 and label the edges with 1,2,…,81,2,\ldots, 8. There are special rules he must follow:
If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd.
Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If NN is the number of possible octagon labelings, find the remainder when NN is divided by 100100.
Proposed by Harry Kim
MOAA 2023
2023 MOAA Team P8

Source:

10/14/2023
Two consecutive positive integers nn and n+1n+1 have the property that they both have 66 divisors but a different number of distinct prime factors. Find the sum of the possible values of nn.
Proposed by Harry Kim
MOAA 2023
2023 MOAA Gunga P8

Source:

10/15/2023
Let ABCDABCD be a parallelogram with area 160. Let diagonals ACAC and BDBD intersect at EE. Point PP is on AE‾\overline{AE} such that EC=4EPEC = 4EP. If line DPDP intersects ABAB at FF, find the area of BFPCBFPC.
Proposed by Andy Xu
MOAA 2023