MathDB

1

Part of 2019 Saint Petersburg Mathematical Olympiad

Problems(3)

2019 Saint Petersburg Grade 11 P1

Source: 2019 Saint Petersburg

4/14/2019
A polynomial f(x)f(x) of degree 20002000 is given. It's known that f(x2−1)f(x^2-1) has exactly 34003400 real roots while f(1−x2)f(1-x^2) has exactly 27002700 real roots. Prove that there exist two real roots of f(x)f(x) such that the difference between them is less that 0.0020.002.
(А. Солынин)
Thanks to the user Vlados021 for translating the problem.
algebra
non-constant arithmetic progression a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}

Source: St. Petersburg 2019 10.1

5/1/2019
For a non-constant arithmetic progression (an)(a_n) there exists a natural nn such that an+an+1=a1+…+a3n−1a_{n}+a_{n+1} = a_{1}+…+a_{3n-1} . Prove that there are no zero terms in this progression.
arithmetic sequencealgebraArithmetic ProgressionRecurrencerecurrence relation
1 palindrome between 2 squares of numbers with 1001 digits

Source: St. Petersburg 2019 9.1

5/2/2019
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have 10011001 digits. Prove that strictly between these squares, there is one palindrome.
number theorypalindromesDigits