MathDB

p1. A quadratic equation has the natural roots

a

and

b

. Another quadratic equation has roots

b

and

c

with

a\ne c

. If

a

b

, and

c

are prime numbers less than

15

, how many triplets

(a,b,c)

that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to

1

)?
p2. In Indonesia, was formerly known the "Archipelago Fraction''. The Archipelago Fraction is a fraction

\frac{a}{b}

such that

a

and

b

are natural numbers with

a < b

. Find the sum of all Archipelago Fractions starting from a fraction with

b = 2

b = 1000

.
p3. Look at the following picture. The letters

a, b, c, d

, and

e

in the box will replaced with numbers from

1, 2, 3, 4, 5, 6, 7, 8

, or

9

, provided that

a,b, c, d

, and

e

must be different. If it is known that

ae = bd

, how many arrangements are there? https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png
p4. Given a triangle

ABC

with

A

as the vertex and

BC

as the base. Point

P

lies on the side

CA

. From point

A

a line parallel to

PB

is drawn and intersects extension of the base at point

D

. Point

E

lies on the base so that

CE : ED = 2 :3

. If

F

is the midpoint between

E

and

C

, and the area of triangle ABC is equal with

35

^2

, what is the area of triangle

PEF

?
p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to

1001

, find the sum of all the numbers written on the sides of the cube.

day 1

Problems(1)