Combinatorics problem in IMO 2013

combinatoric problem in international mathematic olympiad ( IMO) 2013

Problem 1:

Let n be a positive integer. Find the smallest integer k with the following property: Given any real numbers a1, . . . , ad such that a1 + a2 + … + ad = n and 0\leqslant a_i\leqslant 1 for i = 1 , 2, …d. it is possible to partition these numbers into k groups (some of which may be empty) such that the sum of the numbers in each group is at most 1.

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Also Read : Combinatoric problem in IMO 2014

Problem 2 :

In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw k lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value of k such that the goal is attainable for every possible configuration of 4027 points.

Problem 3:

A crazy physicist discovered a new kind of particle which he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.

  • If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
  • At any moment, he may double the whole family of imons in his lab by creating a copy I’ of each imon I. During this procedure, the two copies I’ and J’ become entangled if and only if the original imons I and J are entangled, and each copy I’ becomes entangled with its original imon I; no other entanglements occur or disappear at this moment.

Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

Problem 4 :

Let n be a positive integer, and let A be a subset of {1, . . . ,n}. An A-partition of n into k parts is a representation of n as a sum n = a1 + . . . + ak, where the parts a belong to A and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set { a1 , . . . , ak }.

We say that an A-partition of n into k parts is optimal if there is no A-partition of n into r parts with r < k. Prove that any optimal A-partition of n contains at most \sqrt[3]{6n} different parts.

Problem 5:

Let r be a positive integer, and let a0 , a1, . . . . be an infinite sequence of real numbers. Assume that for all nonnegative integers m and s there exists a positive integer n\in [m+1,m+r] such that

Prove that the sequence is periodic, i. e. there exists some p\geqslant 1 such that an+p = an for all n\geqslant 0.

Problem 6:

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of fights. The distance between two cities is defined to be the least possible number of fights required to go from one of them to the other. It is known that for any city there are at most 100 cities at distance exactly three from it. Prove that there is no city such that more than 2550 other cities have distance exactly four from it.

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