MathDB

The problems are really difficult to find online, so here are the problems.
P1. It is known that two positive integers

m

and

n

satisfy

10n - 9m = 7

dan

m \leq 2018

. The number

k = 20 - \frac{18m}{n}

is a fraction in its simplest form. a) Determine the smallest possible value of

k

. b) If the denominator of the smallest value of

k

is (equal to some number)

N

, determine all positive factors of

N

. c) On taking one factor out of all the mentioned positive factors of

N

above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.
I added this because my translation is a bit weird. [hide=Indonesian Version] Diketahui dua bilangan bulat positif

m

dan

n

dengan

10n - 9m = 7

dan

m \leq 2018

. Bilangan

k = 20 - \frac{18m}{n}

merupakan suatu pecahan sederhana. a) Tentukan bilangan

k

terkecil yang mungkin. b) Jika penyebut bilangan

k

terkecil tersebut adalah

N

, tentukan semua faktor positif dari

N

. c) Pada pengambilan satu faktor dari faktor-faktor positif

N

di atas, tentukan peluang terambilnya satu faktor kelipatan 4.
P2. Let the functions

f, g : \mathbb{R} \to \mathbb{R}

be given in the following graphs. [hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description: For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with

X

Y

depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their

X

Y

coordinates only (since their other coordinates are definitely 0). Graph (1) is the function

f

, who is a quadratic function with -2 and 4 as its

X

-intercepts and 4 as its

Y

-intercept. You also put

f

right besides the curve you have, preferably just on the right-up direction of said curve. Graph (2) is the function

g

, which is piecewise. For

x \geq 0

g(x) = \frac{1}{2}x - 2

, whereas for

x < 0

g(x) = - x - 2

. You also put

g

right besides the curve you have, on the lower right of the line, on approximately

x = 2

. Define the function

g \circ f

with

(g \circ f)(x) = g(f(x))

for all

x \in D_f

where

D_f

is the domain of

f

. a) Draw the graph of the function

g \circ f

. b) Determine all values of

x

so that

-\frac{1}{2} \leq (g \circ f)(x) \leq 6

.
P3. The quadrilateral

ABCD

has side lengths

AB = BC = 4\sqrt{3}

cm and

CD = DA = 4

cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral

ABCD

.
P4. There exists positive integers

x

and

y

, with

x < 100

and

y > 9

. It is known that

y = \frac{p}{777} x

, where

p

is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of

y

.
P5. The 8-digit number

\overline{abcdefgh}

(the original problem does not have an overline, which I fixed) is arranged from the set

\{1, 2, 3, 4, 5, 6, 7, 8\}

. Such number satisfies

a + c + e + g \geq b + d + f + h

. Determine the quantity of different possible (such) numbers.

Problem Statement