MathDB

p1. It is known that

a_1=2

a_2=3

. For

k > 2

, define

a_k=\frac{1}{2}a_{k-2}+\frac{1}{3}a_{k-1}

. Find the infinite sum of of

a_1+a_2+a_3+...

p2. The multiplied number is a natural number in two-digit form followed by the result time. For example,

7\times 8 = 56

, then

7856

and

8756

are multiplied numbers .

2\times 3 = 6

, then

236

and

326

are multiplied.

2\times 0 = 0

, then

200

is the multiplied. For the record, the first digit of the number times can't be

0

. a. What is the difference between the largest and the smallest multiplied number? b. Find all the multiplied numbers that consist of three digits and each digit is square number. c. Given the following "boxes" that must be filled with multiple numbers. https://cdn.artofproblemsolving.com/attachments/b/6/ac086a3d1a0549fae909c072224605430daf1d.png Determine the contents of the shaded box. Is this content the only one? d. Complete all the empty boxes above with multiplied numbers.
p3. Look at the picture of the arrangement of three squares below. https://cdn.artofproblemsolving.com/attachments/1/3/c0200abae77cc73260b083117bf4bafc707eea.pngProve that

\angle BAX + \angle CAX = 45^o

p4. Prove that

(n-1)n (n^3 + 1)

is always divisible by

6

for all natural number

n

Problem Statement