MathDB

2021 Alg/NT Div 1 P7

Source:

3/2/2021

As a gift, Dilhan was given the number

n=1^1\cdot2^2\cdots2021^{2021}

, and each day, he has been dividing

n

by

2021!

exactly once. One day, when he did this, he discovered that, for the first time,

n

was no longer an integer, but instead a reduced fraction of the form

\frac{a}b

. What is the sum of all distinct prime factors of

b

?
Proposed by Adam Bertelli

number theoryalgebra

2021 Geo Div 1 P7

Source:

3/2/2021

Convex pentagon

ABCDE

has

\overline{BC}=17

,

\overline{AB}=2\overline{CD}

, and

\angle E=90^\circ

. Additionally,

\overline{BD}-\overline{CD}=\overline{AC}

, and

\overline{BD}+\overline{CD}=25

. Let

M

and

N

be the midpoints of

BC

and

AD

respectively. Ray

EA

is extended out to point

P

, and a line parallel to

AD

is drawn through

P

, intersecting line

EM

at

Q

. Let

G

be the midpoint of

AQ

. Given that

N

and

G

lie on

EM

and

PM

respectively, and the perimeter of

\triangle QBC

is

42

, find the length of

\overline{EM}

.
Proposed by Adam Bertelli

geometry

2021 Combo Div 1 P7

Source:

3/2/2021

How many non-decreasing tuples of integers

(a_1, a_2, \dots, a_{16})

are there such that

0 \leq a_i \leq 16

for all

i

, and the sum of all

a_i

is even?
Proposed by Nancy Kuang

combinatorics

1.7

Problems(3)