MathDB

1

Part of 1999 China National Olympiad

Problems(2)

F is orthocentre of ABC iff HD||CF and H is on (ABC)

Source: Chinese MO 1999

3/19/2012
Let ABCABC be an acute triangle with ∠C>∠B\angle C>\angle B. Let DD be a point on BCBC such that ∠ADB\angle ADB is obtuse, and let HH be the orthocentre of triangle ABDABD. Suppose that FF is a point inside triangle ABCABC that is on the circumcircle of triangle ABDABD. Prove that FF is the orthocenter of triangle ABCABC if and only if HD∣∣CFHD||CF and HH is on the circumcircle of triangle ABCABC.
geometrycircumcirclegeometry proposed
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Source: Chinese MO 1999

10/12/2005
Let mm be a positive integer. Prove that there are integers a,b,ka, b, k, such that both aa and bb are odd, k≥0k\geq0 and 2m=a19+b99+k⋅219992m=a^{19}+b^{99}+k\cdot2^{1999}
modular arithmeticnumber theory unsolvednumber theory