MathDB

p1. Let

U_n

be a sequence of numbers that satisfy:

U_1=1

U_n=1+U_1U_2U_3...U_{n-1}

for

n=2,3,...,2020

Prove that

\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2

p2. If

a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil

b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor

, and

c=a-b

, then determine the value of

c

.

p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks

2

fish, then the probability that the

2

fish are of the same sex is

1/2

. Prove that the number of koi fish in the aquarium is a perfect square.
p4. A pharmacist wants to put

155

ml of liquid into

3

bottles. There are 3 bottle choices, namely
a. Bottle A

\bullet

Capacity:

5

\bullet

The price of one bottle is

10,000

\bullet

If you buy the next bottle, you will get a

20\%

discount, up to the

4

th purchase or if you buy

4

bottles, get

1

free bottle A
b. Bottle B

\bullet

Capacity:

8

\bullet

The price of one bottle is

15.000

\bullet

If you buy

2

20\%

discount

\bullet

If you buy

3

: Free

1

bottle of B
c. Bottle C

\bullet

Capacity :

14

\bullet

Buy

1

25.000

\bullet

Buy

2

: Free

1

bottle of A

\bullet

Buy

3

: Free

1

bottle of B
If in one purchase, you can only buy a maximum of

4

bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity).
p5. Two circles, let's say

L_1

and

L_2

have the same center, namely at point

O

. Radius of

L_1

10

cm and radius of

L_2

5

cm. The points

A, B, C, D, E, F

lie on

L_1

so the arcs

AB,BC,CD,DE,EF,FA

are equal. The points

P, Q, R

lie on

L_2

so that the arcs

PQ,QR,RS

are equal and

PA=PF=QB=QC=RD=RD

. Determine the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png

Problem Statement