MathDB

3

Part of 1998 IMO Shortlist

Problems(3)

IMO ShortList 1998, number theory problem 3

Source: IMO ShortList 1998, number theory problem 3

10/22/2004
Determine the smallest integer n≥4n\geq 4 for which one can choose four different numbers a,b,ca,b,c and dd from any nn distinct integers such that a+b−c−da+b-c-d is divisible by 2020.
modular arithmeticpigeonhole principlenumber theoryDivisibilityIMO Shortlistcombinatorics
IMO ShortList 1998, algebra problem 3

Source: IMO ShortList 1998, algebra problem 3

10/22/2004
Let x,yx,y and zz be positive real numbers such that xyz=1xyz=1. Prove that x3(1+y)(1+z)+y3(1+z)(1+x)+z3(1+x)(1+y)≥34. \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.
inequalitiesrearrangement inequality3-variable inequalityIMO ShortlistalgebraHigh School Olympiads
IMO ShortList 1998, combinatorics theory problem 3

Source: IMO ShortList 1998, combinatorics theory problem 3

10/22/2004
Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 9 1 6 5 3‾\underline{6\ 5\ 3} 2 7 4 82\ 7\ 4\ 8 may be changed to 919 1 3 5 6‾\underline{3\ 5\ 6} 2 7 4 82\ 7\ 4\ 8. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in ascending or descending order.
combinatoricspermutationIMO Shortlist