4

Part of 2014 Czech-Polish-Slovak Junior Match

Problems(2)

find smallest t such that 11 | a_t , where a_n=12345678091011...n

Source: Czech-Polish-Slovak Junior Match 2014, Individual p4 CPSJ

a_n

is formed by writing in succession, without spaces, the numbers

1, 2, ..., n

a_{11} = 1234567891011

). Find the smallest number t such that

11 | a_t

number theorydividesdivisibleDigits

perpendicular wanted when 2 triangles have same areas, circle related

Source: Czech-Polish-Slovak Junior Match 2014, Team p4 CPSJ

M

is the midpoint of the side

AB

of an acute triangle

ABC

. Circle with center

M

passing through point

C

, intersects lines

AC ,BC

for the second time at points

P,Q

respectively. Point

R

lies on segment

AB

such that the triangles

APR

BQR

have equal areas. Prove that lines

PQ

CR

are perpendicular.

area of a trianglegeometryperpendicularcircleareasequal areas