Math Team Mixed Problems in Combinatorics and Number Theory

1 Warm-ups

1. Show that if 2n+ 1 real numbers have the property that the sum of any n is less than the sumof the remaining n+ 1, then all these numbers are positive.

2. Prove that the product of four consecutive positive integers cannot be a perfect square.

3. How many permutations of the 26 letters of the English alphabet do not contain any of thestrings fish, rat or bird?

2 Problems

1. Show that every positive rational number can be written as a quotient of products of factorialsof (not necessarily distinct) primes.

2. Find the positive integer solutions to 3x + 4y = 5z

3. What is the maximum number of rational points that can lie on a circle in R2 whose centeris not a rational point? (A rational point is a point both of whose coordinates are rationalnumbers).

4. Solve in positive integers the following equation: xx+y = yyx

5. Find all positive integers n for which 1n + 9n + 10n = 5n + 6n + 11n

6. In a finite sequence of real numbers the sum of any seven successive terms is negative, and thesum of any elven successive terms is positive. Determine the maximum number of terms in thesequence.

7. A sheet of paper in the shape of a square is cut by a line into two pieces. One of the pieces iscut again by a line, and so on. What is the minimum number of cuts one should perform suchthat among the pieces one can find one hundred polygons with twenty sides ?

3 Challenge

1. A round-robin tournament of 2n teams lasted for 2n 1 days, as follows. On each day, everyteam played one game against another team, with one team winning and one team losing ineach of the n games. Over the course of the tournament, each team played every other teamexactly once. Can one necessarily choose one winning team from each day without choosingany team more than once?

2. Let n > 1 be a natural number. Let U = {1, 2, ..., n}, and define A B to be the set of allthose elements of U which belong to exactly one of A and B. Show that || 2n1, where is a collection of subsets of U such that for any two distinct elements of A,B of we have|AB| 2. Also find all such collections for which the maximum is attained.

3. Find all integers n for which there exists an equiangular n-gon whose side lengths are distinctrational numbers.

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