MathDB

1

Counting messed up number theory messed up P6

n=(q + 2)q^{2021}

where

q=10^9+7

. For every

k<=n

and prime

p|n

, define

f_{p,k}(n)

=

v_{p}

(\binom{n}{k})

(

v_{p}

(i)

is the highest power of

p

that divides

i

). Let

m

be the maximum possible (over all

k

) value of the expression

\prod_{p\text{,prime,} p|n} f_{p,k}

. Find the sum of the digits of

m

.

5

1

wHeN cOmBiNaToRiAl sToRiEs MeSs WiTh GeOmEtRy

2021

copies of each of the number from

1

to

5

are initially written on the board.Every second Alice picks any two f these numbers, say

a

and

b

and writes

\frac{ab}{c}

.Where

c

is the length of the hypoteneus with sides

a

and

b

.Alice stops when only one number is left.If the minnimum number she could write was

x

and the maximum number she could write was

y

then find the greatest integer lesser than

2021^2xy

.
Does any body know how to use floors and ceiling function?cuz actuall formation used ceiling,but since Idk how to use ceiling I had to do it like this :(]

1

Probability and Expected value, flash back to the AMC

We have

2022

1s

written on a board in a line. We randomly choose a strictly increasing sequence from

{1, 2, . . . , 2022}

such that the last term is

2022

. If the chosen sequence is

a_1, a_2, ..., a_k

(

k

is not fixed), then at the

i^{th}

step, we choose the first a

_i

numbers on the line and change the 1s to 0s and 0s to 1s. After

k

steps are over, we calculate the sum of the numbers on the board, say

S

. The expected value of

S

is

\frac{a}{b}

where

a, b

are relatively prime positive integers. Find

a + b.

4

1

Easy Geometry

Given

\triangle ABC

with

\angle A = 15^{\circ}

, let

M

be midpoint of

BC

and let

E

and

F

be points on ray

BA

and

CA

respectively such that

BE = BM = CF

. Let

R_1

be the radius of

(MEF)

and

R_{2}

be radius of

(AEF)

. If

\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}

where

a,b,c

are integers. Find

a^{b^{c}}

3

1

Valid paths and a whole lot of scary notation

We call a path Valid if i. It only comprises of the following kind of steps: A.

(x, y) \rightarrow (x + 1, y + 1)

B.

(x, y) \rightarrow (x + 1, y - 1)

ii. It never goes below the x-axis. Let

M(n)

= set of all valid paths from

(0,0)

, to

(2n,0)

, where

n

is a natural number. Consider a Valid path

T \in M(n)

. Denote

\phi(T) = \prod_{i=1}^{2n} \mu_i

, where

\mu_i

= a)

1

, if the

i^{th}

step is

(x, y) \rightarrow (x + 1, y + 1)

b)

y

, if the

i^{th}

step is

(x, y) \rightarrow (x + 1, y - 1)

Now Let

f(n) =\sum _{T \in M(n)} \phi(T)

. Evaluate the number of zeroes at the end in the decimal expansion of

f(2021)

2

1

tRiCkY NuMbEr ThEoRy

Define a function

g :\mathbb{N} \rightarrow \mathbb{R}

Such that

g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}

. Find the last 2 digits in the decimal representation of

g(2021)

.

STEMS 2022 Math Cat A Qualifier Round

Subcontests

Counting messed up number theory messed up P6

wHeN cOmBiNaToRiAl sToRiEs MeSs WiTh GeOmEtRy

Probability and Expected value, flash back to the AMC

Easy Geometry

Valid paths and a whole lot of scary notation

tRiCkY NuMbEr ThEoRy