MathDB

p4 Define the sequence

{a_n}

as follows: 1)

a_1 = -1

, and 2) for all

n \ge 2

a_n = 1 + 2 + . . . + n - (n + 1)

. For example,

a_3 = 1+2+3-4 = 2

. Find the largest possible value of

k

such that

a_k+a_{k+1} = a_{k+2}

.
p5 The taxicab distance between two points

(a, b)

and

(c, d)

on the coordinate plane is

|c-a|+|d-b|

. Given that the taxicab distance between points

A

and

B

8

and that the length of

AB

k

, find the minimum possible value of

k^2

.
p6 For any two-digit positive integer

\overline{AB}

, let

f(\overline{AB}) = \overline{AB}-A\cdot B

, or in other words, the result of subtracting the product of its digits from the integer itself. For example,

f(\overline{72}) = 72-7\cdot 2 = 58

. Find the maximum possible

n

such that there exist distinct two-digit integers

\overline{XY}

and

\overline{WZ}

such that

f(\overline{XY} ) = f(\overline{WZ}) = n

Problem Statement