MathDB

4

Part of 2009 Tuymaada Olympiad

Problems(4)

24-element subset B containing neither of the sets A_1,...

Source: Tuymaada 2009, Junior League, First Day, Problem 4

7/19/2009
Each of the subsets A1 A_1, A2 A_2, …, \dots, An A_n of a 2009-element set X X contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in X X there is a 24-element subset B B containing neither of the sets A1 A_1, A2 A_2, …, \dots, An A_n.
algebrapolynomialcalculusderivativefloor functioncombinatorics unsolvedcombinatorics
The sum of several numbers is not greater than 200

Source: Tuymaada 2009, Junior League, Second Day, Problem 4

7/19/2009
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. Proposed by A. Khabrov
floor functioninequalitiesfunctionalgebra unsolvedalgebra
Is there a positive integer n such that among 200th digits

Source: Tuymaada 2009, Senior League, First Day, Problem 4

7/19/2009
Is there a positive integer n n such that among 200th digits after decimal point in the decimal representations of n \sqrt{n}, \sqrt{n\plus{}1}, \sqrt{n\plus{}2}, …, \ldots, \sqrt{n\plus{}999} every digit occurs 100 times? Proposed by A. Golovanov
algebra unsolvedalgebra
Every polynomial P(x) of degree 99

Source: Tuymaada 2009, Senior League, Second Day, Problem 4

7/19/2009
Determine the maximum number h h satisfying the following condition: for every a∈[0,h] a\in [0,h] and every polynomial P(x) P(x) of degree 99 such that P(0)\equal{}P(1)\equal{}0, there exist x1,x2∈[0,1] x_1,x_2\in [0,1] such that P(x_1)\equal{}P(x_2) and x_2\minus{}x_1\equal{}a. Proposed by F. Petrov, D. Rostovsky, A. Khrabrov
algebrapolynomialfloor functionfunctional equationalgebra unsolved