MathDB

p1. Is there any natural number n such that

n^2 + 5n + 1

is divisible by

49

? Explain.
p2. It is known that the parabola

y = ax^2 + bx + c

passes through the points

(-3,4)

and

(3,16)

, and does not cut the

x

-axis. Find all possible abscissa values for the vertex point of the parabola.
p3. It is known that

T.ABC

is a regular triangular pyramid with side lengths of

2

cm. The points

P, Q, R

, and

S

are the centroids of triangles

ABC

TAB

TBC

and

TCA

, respectively . Determine the volume of the triangular pyramid

P.QRS

.
p4. At an event invited

13

special guests consisting of

8

people men and

5

women. Especially for all those special guests provided

13

seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests.
p5. A table of size

n

rows and

n

columns will be filled with numbers

1

-1

so that the product of all the numbers in each row and the product of all the numbers in each column is

-1

. How many different ways to fill the table?

Problem Statement