MathDB

Problem 3

Part of 2001 Federal Math Competition of S&M

Problems(3)

Prime number ineq

Source: Serbian national cometition

4/11/2014
Let p1,p2,...,pnp_{1}, p_{2},...,p_{n}, where n>2n>2, be the first nn prime numbers. Prove that 1p12+1p22+...+1pn2+1p1p2...pn<12\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}
inequalitiesinequalities unsolved
{0,1,...,2k+1}, odd zeroes

Source: Serbia &amp; Montenegro 2001 3,4th Grade P3

6/2/2021
Let kk be a positive integer and NkN_k be the number of sequences of length 20012001, all members of which are elements of the set {0,1,2,…,2k+1}\{0,1,2,\ldots,2k+1\}, and the number of zeroes among these is odd. Find the greatest power of 22 which divides NkN_k.
combinatorics
paint in plane

Source: Serbia 2001

4/28/2008
Determine all positive integers n n for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly n n colors are used. (iii) Each line is painted in at most two different colors.
combinatorics unsolvedcombinatorics