MathDB
Indonesia Regional MO 2008 Part B

Source:

October 2, 2021
algebrageometrycombinatoricsnumber theoryIndonesia Regional MO

Problem Statement

p1. Find all pairs of natural numbers (x,n)(x, n) that satisfy 1+x+x2+...+xn=401 + x + x^2 + ... + x^n = 40
p2. Given a real polynomial P(x)=x2008+a1x2007+a2x2006+...+a2007x+a2008P(x) = x^{2008} + a_1x^{2007} + a_2x^{2006} + ...+ a_{2007}x + a_{2008} and Q(x)=x2+2x+2008Q(x) = x^2 + 2x + 2008. Suppose the equation P(x)=0P(x) = 0 has 20082008 real solutions and P(2008)≤1P(2008)\le 1. Show that the equation P(Q(x))=0P(Q(x)) = 0 has a real solution.
[url=https://artofproblemsolving.com/community/c6h211746p1167343]p3. The inscribed circle of triangle ABC, touches the sides BCBC, CACA, and ABAB at DD, EE, and FF, respectively. Through DD, a perpendicular line EFEF is drawn that intersects EFEF at GG. Prove that \frac{FG}{EG}=\frac{BF}{CE}.
p4. The numbers 1,2,3,...,91, 2, 3, ..., 9 are arranged in a circle randomly. Prove that there are three adjacent numbers whose sum is greater than 1515.
5. Determine the number of positive 55-digit palindromes that are divisible by 33. A palindrome is a number/word that is the same if read from left to right or vice versa. For example, 3535335353 is a palindrome, while 1424214242 is not.
Back to ProblemsView on AoPS