MathDB

7

Part of 1999 IMO Shortlist

Problems(2)

IMO ShortList 1999, geometry problem 7

Source: IMO ShortList 1999, geometry problem 7

11/13/2004
The point MM is inside the convex quadrilateral ABCDABCD, such that MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD}   \textnormal{and}   \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. Prove that AB⋅CM=BC⋅MDAB \cdot CM = BC \cdot MD and BM⋅AD=MA⋅CD.BM \cdot AD = MA \cdot CD.
geometrycircumcirclereflectionsimilar trianglesquadrilateralIMO Shortlist
IMO ShortList 1999, combinatorics problem 7

Source: IMO ShortList 1999, combinatorics problem 7

11/14/2004
Let p>3p >3 be a prime number. For each nonempty subset TT of {0,1,2,3,…,p−1}\{0,1,2,3, \ldots , p-1\}, let E(T)E(T) be the set of all (p−1)(p-1)-tuples (x1,…,xp−1)(x_1, \ldots ,x_{p-1} ), where each xi∈Tx_i \in T and x1+2x2+…+(p−1)xp−1x_1+2x_2+ \ldots + (p-1)x_{p-1} is divisible by pp and let ∣E(T)∣|E(T)| denote the number of elements in E(T)E(T). Prove that ∣E({0,1,3})∣≥∣E({0,1,2})∣|E(\{0,1,3\})| \geq |E(\{0,1,2\})| with equality if and only if p=5p = 5.
countingSubsetsSet systemsDivisibilitycombinatoricsIMO Shortlistcombinatorial inequality