MathDB

2022 Algebra/NT #10

Source:

3/11/2022

Compute the smallest positive integer

n

for which there are at least two odd primes

p

such that

\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0

. Note: for a prime

p

and a positive integer

m

,

v_p(m)

is the exponent of the largest power of

p

that divides

m

; for example,

v_3(18) = 2

.

number theory

2022 Team 10

Source:

3/14/2022

On a board the following six vectors are written:

(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1).

Given two vectors

v

and

w

on the board, a move consists of erasing

v

and

w

and replacing them with

\frac{1}{\sqrt2} (v + w)

and

\frac{1}{\sqrt2} (v - w)

. After some number of moves, the sum of the six vectors on the board is

u

. Find, with proof, the maximum possible length of

u

.

Vectorsgeometry

2022 Geometry 10

Source:

3/14/2022

Suppose

\omega

is a circle centered at

O

with radius

8

. Let

AC

and

BD

be perpendicular chords of

\omega

. Let

P

be a point inside quadrilateral

ABCD

such that the circumcircles of triangles

ABP

and

CDP

are tangent, and the circumcircles of triangles

ADP

and

BCP

are tangent. If

AC = 2\sqrt{61}

and

BD = 6\sqrt7

,then

OP

can be expressed as

\sqrt{a}-\sqrt{b}

for positive integers

a

and

b

. Compute

100a + b

.

geometry

2022 Combinatorics 10

Source:

3/18/2022

Let

S

be a set of size

11

. A random

12

-tuple

(s_1, s_2, . . . , s_{12})

of elements of

S

is chosen uniformly at random. Moreover, let

\pi : S \to S

be a permutation of

S

chosen uniformly at random. The probability that

s_{i+1}\ne \pi (s_i)

for all

1 \le i \le 12

(where

s_{13} = s_1

) can be written as

\frac{a}{b}

where

a

and

b

are relatively prime positive integers. Compute

a

.

probabilitycombinatorics

10

Problems(4)