MathDB

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Part of 2015 Caucasus Mathematical Olympiad

Problems(5)

4-digit number so that with his reverse one is divisible by 101

Source: Caucasus 2015 7.1

4/26/2019
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by 101101?
number theoryDigitdivisibleSum
4 numbers so that adding 2 products of pairs you get a prime number

Source: Caucasus 2015 8.1

4/26/2019
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
number theoryprimeNatural NumbersProductprime numbers
max in 10, all either always tell the truth either always tells lies

Source: Caucasus 2015 9.1

4/26/2019
At the round table, 1010 people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ? (A statement that is at least partially false is considered false.)
combinatoricsTrue or False
solve (x-a)(x-b)=(x-c)(x-d) if a+d=b+c=2015, a \ne c

Source: Caucasus 2015 10.1

4/26/2019
Find the roots of the equation (x−a)(x−b)=(x−c)(x−d)(x-a)(x-b)=(x-c)(x-d), if you know that a+d=b+c=2015a+d=b+c=2015 and a≠ca \ne c (numbers a,b,c,da, b, c, d are not given).
algebraequation
exists 8-digit number where divided by 1st digit gives remainder 1, etc ?

Source: Caucasus 2018 11.1

4/26/2019
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder 11, when divided by the second digit will give the remainder 22, ..., when divided by the eighth digit will give the remainder 88?
number theoryDigitsDigitdivisibleremainder