MathDB

4

Part of 1998 IMO Shortlist

Problems(4)

IMO ShortList 1998, geometry problem 4

Source: IMO ShortList 1998, geometry problem 4

10/22/2004
Let M M and N N be two points inside triangle ABC ABC such that \angle MAB \equal{} \angle NAC  \mbox{and}  \angle MBA \equal{} \angle NBC. Prove that \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
geometryreflectiontrigonometrycircumcircleIMO ShortlistIsogonal conjugate
IMO ShortList 1998, number theory problem 4

Source: IMO ShortList 1998, number theory problem 4

10/22/2004
A sequence of integers a1,a2,a3,… a_{1},a_{2},a_{3},\ldots is defined as follows: a_{1} \equal{} 1 and for n≥1 n\geq 1, a_{n \plus{} 1} is the smallest integer greater than an a_{n} such that a_{i} \plus{} a_{j}\neq 3a_{k} for any i,j i,j and k k in \{1,2,3,\ldots ,n \plus{} 1\}, not necessarily distinct. Determine a1998 a_{1998}.
number theoryInteger sequenceCalculateIMO Shortlist
IMO ShortList 1998, algebra problem 4

Source: IMO ShortList 1998, algebra problem 4; Polish 1st round, 1999

10/22/2004
For any two nonnegative integers nn and kk satisfying n≥kn\geq k, we define the number c(n,k)c(n,k) as follows: - c(n,0)=c(n,n)=1c\left(n,0\right)=c\left(n,n\right)=1 for all n≥0n\geq 0; - c(n+1,k)=2kc(n,k)+c(n,k−1)c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right) for n≥k≥1n\geq k\geq 1. Prove that c(n,k)=c(n,n−k)c\left(n,k\right)=c\left(n,n-k\right) for all n≥k≥0n\geq k\geq 0.
functioncombinatoricscountingsymmetrybinomial coefficientsIMO Shortlist
IMO ShortList 1998, combinatorics theory problem 4

Source: IMO ShortList 1998, combinatorics theory problem 4

10/22/2004
Let U={1,2,…,n}U=\{1,2,\ldots ,n\}, where n≥3n\geq 3. A subset SS of UU is said to be split by an arrangement of the elements of UU if an element not in SS occurs in the arrangement somewhere between two elements of SS. For example, 13542 splits {1,2,3}\{1,2,3\} but not {3,4,5}\{3,4,5\}. Prove that for any n−2n-2 subsets of UU, each containing at least 2 and at most n−1n-1 elements, there is an arrangement of the elements of UU which splits all of them.
combinatoricsSet systemsSubsetsIMO Shortlist