MathDB

p1. The parabola

y = ax^2 + bx

a < 0

, has a vertex

C

and intersects the

x

-axis at different points

A

and

B

. The line

y = ax

intersects the parabola at different points

A

and

D

. If the area of triangle

ABC

is equal to

|ab|

times the area of triangle

ABD

, find the value of

b

in terms of

a

without use the absolute value sign.
p2. It is known that

a

is a prime number and

k

is a positive integer. If

\sqrt{k^2-ak}

is a positive integer, find the value of

k

in terms of

a

.
p3. There are five distinct points,

T_1

T_2

T_3

T_4

, and

T

on a circle

\Omega

. Let

t_{ij}

be the distance from the point

T

to the line

T_iT_j

or its extension. Prove that

\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}

and

\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}

https://cdn.artofproblemsolving.com/attachments/2/8/07fff0a36a80708d6f6ec6708f609d080b44a2.png

p4. Given a

7

-digit positive integer sequence

a_1, a_2, a_3, ..., a_{2017}

with

a_1 < a_2 < a_3 < ...<a_{2017}

. Each of these terms has constituent numbers in non-increasing order. Is known that

a_1 = 1000000

and

a_{n+1}

is the smallest possible number that is greater than

a_n

. As For example, we get

a_2 = 1100000

and

a_3 = 1110000

. Determine

a_{2017}

.
p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume

V

. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for

a

hours. Then, charging continues using only pump-1 for

b

hours and continues again using only pump-2 for

c

hours. If the operating cost of pump-1 is

15(a + b)

thousand per hour and pump-2 operating cost is

4(a + c)

thousand per hour, determine

b

and

c

so that the operating costs of all pumps are minimum (express

b

and

c

in terms of

a

). Also determine the possible values of

a

Problem Statement