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: $5$ ml $\bullet$ The price of one bottle is $10,000$ Rp $\bullet$ 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: $8$ ml $\bullet$ The price of one bottle is $15.000$ Rp $\bullet$ If you buy $2$ : $20\%$ discount $\bullet$ If you buy $3$ : Free $ 1$ bottle of B c. Bottle C $\bullet$ Capacity : $14$ ml $\bullet$ Buy $ 1$ : $25.000$ Rp $\bullet$ 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 $L_1$ is $10$ cm and radius of $L_2$ is $5$ 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.