Solution to the problem #5834 by E. Usov from the website of K. Polyakov.

task-5834.py

# Helpful hashmaps.
HEX_DEC = dict(zip('0123456789abcdef', tuple(range(16))))  # Pairs of digits - `hexadecimal`: `decimal`
DEC_HEX = dict(zip(tuple(range(16)), '0123456789abcdef'))  # Pairs of digits - `decimal`: `hexadecimal`


# Implementation of the algorithm described in the problem.
def detective(n: int) -> str:
    hex_n = hex(n)[2:]

    hex_n += '0' if n % 2 else 'f'

    for _ in range(2):
        hex_n += DEC_HEX[sum(HEX_DEC[x] for x in hex_n) % 16]

    return hex_n


# The main program.
number = 1
while True:
    tmp = detective(number)

    if tmp.count(max(tmp, key=HEX_DEC.get)) == tmp.count(min(tmp, key=HEX_DEC.get)) * 5:
        print(number)
        break

    number += 1