3

Part of 2000 China Team Selection Test

Problems(2)

Sum of squares of digits of k in base a representation

Source: China TST 2000, problem 3

For positive integer

a \geq 2

N_a

as the number of positive integer

k

with the following property: the sum of squares of digits of

k

in base a representation equals

k

. Prove that: a.)

N_a

is odd; b.) For every positive integer

M

, there exist a positive integer

a \geq 2

N_a \geq M

number theory unsolvednumber theory

Explicit value of N(4)

Source: China TST 2000, problem 6

n

be a positive integer. Denote

M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}

. Define function

f

M

with the following properties: a.)

f(x, y)

takes non-negative integer value; b.)

\sum^n_{y=1} f(x, y) = n - 1

for 1 \eq x \leq n; c.) If

f(x_1, y_1)f(x2, y2) > 0

(x_1 - x_2)(y_1 - y_2) \geq 0.

N(n)

, the number of functions

f

that satisfy all the conditions. Give the explicit value of

N(4)

functionLaTeXalgebra unsolvedalgebra