p1. A telephone number with 7 digits is called a Beautiful Number if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: 7133719, 7131735, 7130713, 1739317, 5433354. If the numbers are taken from 0,1,2,3,4,5,6,7,8 or 9, but the number the first cannot be 0, how many Beautiful Numbers can there be obtained?
p2. Find the number of natural numbers n such that n3+100 is divisible by n+10
p3. A function f is defined as in the following table.
https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png
Based on the definition of the function f above, then a sequence is defined on the general formula for the terms is as follows: U1=2 and Un+1=f(Un) , for n=1,2,3,...
p4. In a triangle ABC, point D lies on side AB and point E lies on side AC. Prove for the ratio of areas: ABCADE=AB×ACAD×AE
p5. In a chess tournament, a player only plays once with another player. A player scores 1 if he wins, 0 if he loses, and 21 if it's a draw. After the competition ended, it was discovered that 21 of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, 21 of the total score one gets is obtained from playing with 9 other players. How many players are there in the competition? algebrageometrynumber theorycombinatoricsindonesia juniors