1

Part of 2017 Czech-Polish-Slovak Match

Problems(2)

find real so that no square of an integer exists in sequence

Source: Czech-Polish-Slovak Match 2017 day 1 P1

Find all positive real numbers

c

such that there are infinitely many pairs of positive integers

(n,m)

satisfying the following conditions:

n \ge m+c\sqrt{m - 1}+1

and among numbers

n. n+1,.... 2n-m

there is no square of an integer.
(Slovakia)

Perfect Squaresnumber theorypositive integers

collinear points out of 3 circumcirlcles

Source: Czech-Polish-Slovak Match 2017 day 2 P1

{ABC}

be a triangle. Line l is parallel to

{BC}

and it respectively intersects side

{AB}

{D}

{AC}

{E}

, and the circumcircle of the triangle

{ABC}

{F}

{G}

{F,D,E,G}

lie in this order on l. The circumcircles of triangles

{FEB}

{DGC}

intersect at points

{P}

{Q}

. Prove that points

{A, P,Q}

are collinear.
(Slovakia)

geometrycollinear