MathDB

p1. Let

AB

be the diameter of the circle and

P

is a point outside the circle. The lines

PQ

and

PR

are tangent to the circles at points

Q

and

R

. The lines

PH

is perpendicular on line

AB

H

. Line

PH

intersects

AR

S

. If

\angle QPH =40^o

and

\angle QSA =30^o

, find

\angle RPS

.
p2. There is a meeting consisting of

40

seats attended by

16

invited guests. If each invited guest must be limited to at least

1

chair, then determine the number of arrangements.
p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from

1

9

. https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png Across: 1. Composite factor of

1001

3. Non-polyndromic numbers 5.

p\times q^3

, with

p\ne q

and

p,q

primes
Down: 1.

a-1

and

b+1

a\ne b

and

p,q

primes 2. multiple of

9

p^3 \times q

, with

p\ne q

and

p,q

primes
p4. Given a function

f:R \to R

and a function

g:R \to R

, so that it fulfills the following figure: https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png Find the number of values of

x

, such that

(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}

.
p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let

k(m,n)

be the number of small areas created if there are

m

control towers and

n

monitoring towers. a. Find the values of

k(4,1)

k(4,2)

k(4,3)

, and

k(4,4)

b. Find the general formula

k(m,n)

with

m

and

n

natural numbers .

Problem Statement