MathDB

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

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

n^3 + 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:

U_1=2

and

U_{n+1}=f(U_n)

, 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:

\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}

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

\frac12

if it's a draw. After the competition ended, it was discovered that

\frac12

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,

\frac12

of the total score one gets is obtained from playing with

9

other players. How many players are there in the competition?

Problem Statement