MathDB

2

Part of 1998 IMO Shortlist

Problems(4)

IMO ShortList 1998, geometry problem 2

Source: IMO ShortList 1998, geometry problem 2

10/22/2004
Let ABCDABCD be a cyclic quadrilateral. Let EE and FF be variable points on the sides ABAB and CDCD, respectively, such that AE:EB=CF:FDAE:EB=CF:FD. Let PP be the point on the segment EFEF such that PE:PF=AB:CDPE:PF=AB:CD. Prove that the ratio between the areas of triangles APDAPD and BPCBPC does not depend on the choice of EE and FF.
geometrycircumcircletrapezoidratioareaIMO Shortlist
IMO ShortList 1998, number theory problem 2

Source: IMO ShortList 1998, number theory problem 2

10/22/2004
Determine all pairs (a,b)(a,b) of real numbers such that a⌊bn⌋=b⌊an⌋a \lfloor bn \rfloor =b \lfloor an \rfloor for all positive integers nn. (Note that ⌊x⌋\lfloor x\rfloor denotes the greatest integer less than or equal to xx.)
floor functionnumber theoryalgebraic identitiesalgebraIMO Shortlist
IMO ShortList 1998, algebra problem 2

Source: IMO ShortList 1998, algebra problem 2

10/22/2004
Let r1,r2,…,rnr_{1},r_{2},\ldots ,r_{n} be real numbers greater than or equal to 1. Prove that 1r1+1+1r2+1+⋯+1rn+1≥nr1r2⋯rnn+1. \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}.
inequalitiesfunctionalgebran-variable inequalityIMO Shortlist
IMO ShortList 1998, combinatorics theory problem 2

Source: IMO ShortList 1998, combinatorics theory problem 2

10/22/2004
Let nn be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or nn 4.) In each multiplication, one can choose independently whether to multiply by 2 or by nn. A positive integer which cannot be so obtained is said to be unattainable. a.) Prove that if n≥9n\geq 9, there are infinitely many unattainable positive integers. b.) Prove that if n=3n=3, all positive integers except 7 are attainable.
combinatoricsinvariantalgorithmIMO Shortlist