MathDB

HMMT Team 2019/5: Partitioning edges of square grid into Ts

Source:

2/17/2019

Find all positive integers

n

such that the unit segments of an

n \times n

grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

HMMTcombinatorics

HMMT Algebra/NT 2019/5: Next term of arithmetico-geometric sequence

Source:

2/17/2019

Let

a_1, a_2, \dots

be an arithmetic sequence and

b_1, b_2, \dots

be a geometric sequence. Suppose that

a_1 b_1 = 20

,

a_2 b_2 = 19

, and

a_3 b_3 = 14

. Find the greatest possible value of

a_4 b_4

.

HMMTalgebra

HMMT Combinatorics 2019/5: Contessa Doesn't Like This "Game"

Source:

2/17/2019

Contessa is taking a random lattice walk in the plane, starting at

(1,1)

. (In a random lattice walk, one moves up, down, left, or right

1

unit with equal probability at each step.) If she lands on a point of the form

(6m,6n)

for

m,n \in \mathbb{Z}

, she ascends to heaven, but if she lands on a point of the form

(6m+3,6n+3)

for

m,n \in \mathbb{Z}

, she descends to hell. What is the probability she ascends to heaven?

HMMTprobability

HMMT Geometry 2019/5: A-isosceles ABC; reflect C over AB

Source:

2/17/2019

Isosceles triangle

ABC

with

AB = AC

is inscibed is a unit circle

\Omega

with center

O

. Point

D

is the reflection of

C

across

AB

. Given that

DO = \sqrt{3}

, find the area of triangle

ABC

.

HMMTgeometry

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