6

Part of 2016 Tuymaada Olympiad

Problems(2)

Russian geometry problem

Source: Tuymaada 2016, Senior League/ P6 and P7 for juniors

a

b

c

d

0<a \leq b \leq d \leq c

{a+c=b+d}

. Prove that for every internal point

P

of a segment with length

a

this segment is a side of a circumscribed quadrilateral with consecutive sides

a

b

c

d

, such that its incircle contains~

P

existance of odd digital number

Source: Tuymaada 2016. Juniors/P6

Is there a positive integer

N>10^{20}

such that all its decimal digits are odd, the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal, and it is divisible by each 20-digit number obtained from it by deleting digits? (Neither deleted nor remaining digits must be consecutive.)