MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2020 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/q1h2686143p23303737]here
Time: 90 minutes

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Write only the answers to the questions given.

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Some questions can have more than one correct answer. You are asked to provide the most correct or exact answer to a question like this. Scores will only be given to the giver of the most correct or most exact answer.

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The system did NOT allow you to revert to the previous problem if you choose to do so, you were only allowed to answer the current problem, and THEN move on to the next problem, until your time runs out. So each problem is a hit or miss, with no possibility of rechecking.

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Each "easy" question is worth 1 (one) point, "medium" question is worth 1.5 points and "hard" question is worth 2 points each.

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There are 16 problems in this round, 8 of which are easy, 4 are medium and 4 are hard.
You are supposed to fill in the blank boxes using this system.
EASY
Question 1. A number of students participate in an exam which has the following composition of problems:

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The first section consists of 3 problems with 2 choices (either True or False)

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The second section consists of 5 problems with 5 choices (A, B, C, D, E). The minimum number of students who needs to participate in the exam so that there always exists two students with identical answers, albeit on the first or even the second section is ....
Question 2. The number of naturals

n < 800

such that

8

divides

\left \lfloor \frac{n}{5} \right \rfloor

, however

8

does not divide

n

is ....
Question 3. It is given a square with a circumradius of 6 units with

O

as its circumcenter (this means the distance between

O

and the each vertex of the square is 6 units). The square is then rotated

45^{\circ}

clockwise with

O

as its center of rotation. Both squares, before and after the rotation, are combined into one figure (look at the attachment) with a perimeter of

K

and an area of

L

. The value of

\left ( \frac{L}{K} \right )^2

is ....
Question 4. Let

x, y

be positive integers and \begin{align*} A &= \sqrt{\log x}, &B = \sqrt{\log y}, \\ C &= \log \sqrt{x}, &D = \log \sqrt{y}. \end{align*} If it is known that

A, B, C, D

are all integers and

A + B + C + D = 24,

then

xy = 10^n

where

n =

....
Question 5. A sequence of integers

u_1, u_2, u_3, \ldots

satisfies: u_{n+1} - u_n = \begin{cases*} 1, &\textrm{ if $n$ is odd} \\ 2, &\textrm{ if $n$ is even}. \end{cases*} If

u_1 + u_2 + \cdots + u_{20} = 360

, then

u_1 =

....
Question 6. It is known the set

S = \{1, 2, 3, 4\}

. The number of nonempty subset sextuples

A_1, A_2, \ldots, A_6

satisfying all three criteria simultaneously:

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A_1 \cap A_2 = \varnothing

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A_1 \cup A_2 \subseteq A_3

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A_3 \subseteq \cdots \subseteq A_6

is ....
Question 7. On a convex quadrilateral

ABCD

, the equalities

\angle{BAD} = \angle{BCD} = 45^{\circ}

BC = AD = 5

hold, and

BC

is not parallel to

AD

. The perimeter of such quadrilateral can be written as

p + q\sqrt{r}

where

p, q, r

are integers and

r

is squarefree (which means it does not have a square factor other than 1). The value of

p + q + r

is ....
Question 8. If

n

is a natural number such that

4n + 808

and

9n + 1621

are perfect squares, then

n =

....
MEDIUM
Question 9 (This problem is flawed!) It is known the triangle

ABC

and the bisector of

\angle{BAC}

cuts side

BC

at point

D

. The circle centred at

C

goes through

D

and cuts

AD

E

(

D \neq E

), and the circle with centre

A

goes through

E

, cutting

AB

X

(

X \neq A

). It is known that

E

is located inside the triangle

ABC

. If

AB = 15, AD = 9

and

AC = 6

, then

BX =

....
Question 10. Let

H

be the set of all natural numbers which can be written as

\frac{10n^2 + 25}{n+2}

for some natural number

n

. The sum of all of the elements of

H

is ....
Question 11. It is given a prism with its base and top being an

n

-sided regular polygon. Every vertex of the prism (all

2n

of them) are labelled with the number

1

-1

. It is known that for every side (face) of the prism, the product of all the numbers which are labelled on all its vertices is

-1

. The sum of all (natural numbers)

n

where

23 \leq n \leq 54

so that such labelling is possible is equal to ....
Question 12. A polynomial

P(x)

satisfies

P \left (x + \frac{2}{x} \right ) = \frac{x^3 + 1}{x} + \frac{x^3 + 8}{2x^2} + 3.

The value of

P(1)

is ....
HARD
Question 13. (This problem is flawed!) On an obtuse triangle, it is known that the longest altitude is 8 and the length of one of the other two altitudes is 3.
If it is known that the third altitude has a length which is equal to a prime number, the length of the third altitude is ....
Question 14. Let

x

and

y

be positive reals such that

\left ( \frac{x}{5} + \frac{y}{3} \right ) \left ( \frac{5}{x} + \frac{3}{y} \right ) = 139.

If the maximum and minimum values of

\frac{x+y}{\sqrt{xy}}

are

M

and

m

respectively, then the value of

M - m

is ....
Question 15. A cube is placed above ground where 5 of its sides (faces) is coloured white and the other one side (face) is coloured black. In the beginning, the black-coloured face is one of the bases of the cube (in Indonesian, the bases are differentiated into a base (alas) and a top (tutup), so here it's called that the black side of the cube is not an "upright face" (i.e. the faces which are not bases)). Then, the cube is rotated on one of its edges which are on the ground such that the bases change, and this process is repeated 8 times. The probability that the black-coloured face lies on one of the bases is ....
[hide=Original wording]Diberikan suatu kubus yang terletak di atas tanah dengan 5 sisi (muka) berwarna putih dan satu sisi (muka) berwarna hitam. Pada awalnya, sisi berwarna hitam bukan merupakan sisi tegak. Kemudian kubus tersebut diputar pada salah satu rusuk pada alasnya sehingga alasnya berganti, dan diulangi sampai 8 kali.
Peluang bahwa sisi berwarna hitam bukan sisi tegak lagi adalah ....
Question 16. Let

m

be a positive integer. A positive integer

n > 1

is called rep-

m

if there exist natural numbers

x, y, z

such that

x + y + z = m

and

\frac{x}{n-1} = \frac{y}{n} = \frac{z}{n+1}.

It is known that there exists exactly

32

rep-

m

numbers, and one of the numbers is 10. The largest natural number

k

such that

10^k

divides

m

is ....

Problem Statement