MathDB

p1. Nurbaya's rectangular courtyard will be covered by a number of paving blocks in the form of a regular hexagon or its pieces like the picture below. The length of the side of the hexagon is

12

cm. https://cdn.artofproblemsolving.com/attachments/6/1/281345c8ee5b1e80167cc21ad39b825c1d8f7b.png Installation of other paving blocks or pieces thereof so that all fully covered page surface. To cover the entire surface The courtyard of the house required

603

paving blocks. How many paving blocks must be cut into models

A, B, C

, and

D

for the purposes of closing. If

17

pieces of model

A

paving blocks are needed, how many the length and width of Nurbaya's yard? Count how much how many pieces of each model

B, C

, and

D

paving blocks are used.
p2. Given the square

PQRS

. If one side lies on the line

y = 2x - 17

and its two vertices lie on the parabola

y = x^2

, find the maximum area of possible squares

PQRS

.
p3. In the triangular pyramid

T.ABC

, the points

E, F, G

, and

H

lie at , respectively

AB

AC

TC

, and

TB

so that

EA : EB = FA : FC = HB : HT = GC : GT = 2:1

. Determine the ratio of the volumes of the two halves of the divided triangular pyramid by the plane

EFGH

.
p4. We know that

x

is a non-negative integer and

y

is an integer. Define all pair

(x, y)

that satisfy

1 + 2^x + 2^{2x + 1} = y^2

.
p5. The coach of the Indonesian basketball national team will select the players for become a member of the core team. The coach will judge five players

A, B, C, D

and

E

in one simulation (or trial) match with total time

80

minute match. At any time there is only one in five players that is playing. There is no limit to the number of substitutions during the match. Total playing time for each player

A, B

, and

C

are multiples of

5

minutes, while the total playing time of each players

D

and

E

are multiples of

7

minutes. How many ways each player on the field based on total playing time?

Problem Statement