Вопрос № 1055483 - Информатика

Some  cryptographic algorithm requires matching natural numbers that can be represented as a sum of two different prime numbers using more than k ways, thus all the three numbers (the unknown and two summands) should have the same number of digits. For example: number 30 matches k-1 case as it can be represented as follows:  and . Number 32 doesn’t match as it can be represented by only one sum: 19+13.
The quantity of 4-digit  decimals for k= 10 is …

The sequence of natural numbers  from  to  is represented without any spaces between them. The decimal sequence obtained is sorted descending the one-digit numbers corresponding to the decimals. For example for   and  the ordered sequence will have the following representation:
98765211110. For  and  the sequence will contain as many “7” digits as____, the first of which is in position with ordinal number _____.
Give your answer in two numbers separated by a comma. For example for  and  in the sequence there will be 19 digits of 7 beginning from position number 39. Thus the result is written as 19,39

Little Piggy decided to categorize his archive containing a lot of different format files. There are pictures, videos, audio, documents, e-tables, documents templates, Web-pages with different file extensions, text of programs in different programming languages and even unknown content files with .hrj extension. Altogether there are 1001111 files. At first Little Piggy decided to move all the pictures into one folder. Thus he selected 5 filtering masks for file names. Most files will be filtered when using the following mask …
Let’s presume that the number of files of every format is relatively similar.

The agent obtained some information concerning a new confidential algorithm for encrypting governmental communication of the potential enemy. However, having almost failed his mission, he had to eat the secret information and  rinse it down with a glass of Coca-Cola. Upon return to the base the data were retrieved from the stomach and partially restored. The basis of encryption was the conversion of number  into a numerical system with a q based number. The base of a number system was lost. However it was found out that the base of the number system was a natural number less than 100 and the number obtained as a result of  conversion contained two 5, one 7 and one 9. Other numbers were permanently lost.  The base of a number system used to convert the operand  is …

In the long-long warehouse, on the long-long shelf there are some heads of cheese stored. Each cheese head is consistently numbered by natural number beginning from 1. One night the first hamster got into a warehouse and  ate up every second cheese beginning from  the second one.  The following night the second hamster  got into a warehouse and ate every second of the remained cheese heads beginning from the first one. After that three rats got into a warehouse and ate every third of the remained cheese heads, beginning from the third. Then the first hamster got hungry and decided to eat some cheese again so it got into a warehouse again and ate every second cheese, beginning from the second, then the second  Hamster ate up every second cheese, beginning from the first, then three rats again… Eventually there was only one cheese head left. Animals decided to leave it not to reveal their presence. On condition that there originally were 123456 heads of cheese, the number of the last cheese head left was …

Let’s suppose that a palindromic number or numeral palindrome is a number of at least two characters  that remain the same when its digits are reversed. For example number 1221 is a palindrome while number  isn’t.
Let’s call a palindrome2 a Base 10 (decimal) palindrome the Base -2 (binary) representation of which is also a palindrome. For example number 33 is a palidrome2 since its Base-2 (binary) representation (100001) is also a palindrome. In the first million of natural numbers there will be so many as ____ palindrome2.