MathDB

4

Part of 2013 Czech-Polish-Slovak Junior Match

Problems(2)

d | \overline{aabbcc} iff d | \overline{abc} where d is two digit number

Source: Czech-Polish-Slovak Junior Match 2013, Individual p4 CPSJ

3/14/2020
Determine the largest two-digit number dd with the following property: for any six-digit number aabbcc‾\overline{aabbcc} number dd is a divisor of the number aabbcc‾\overline{aabbcc} if and only if the number dd is a divisor of the corresponding three-digit number abc‾\overline{abc}.
Note The numbers a≠0,ba \ne 0, b and cc need not be different.
dividesdivisornumber theoryDigits
angle chasing candidate, <ADB =< PDC wanted, circumcircle related

Source: Czech-Polish-Slovak Junior Match 2013, Team p4 CPSJ

3/21/2020
Let ABCDABCD be a convex quadrilateral with ∠DAB=∠ABC=∠BCD>90o\angle DAB =\angle ABC =\angle BCD > 90^o. The circle circumscribed around the triangle ABCABC intersects the sides ADAD and CDCD at points KK and LL, respectively, different from any vertex of the quadrilateral ABCDABCD . Segments ALAL and CKCK intersect at point PP. Prove that ∠ADB=∠PDC\angle ADB =\angle PDC.
geometrycircumcircleequal anglesAngle Chasing