1955 Kurschak Competition

Part of Kürschák Math Competition

Subcontests

(3)

1

only one lattice point in triangle with lattice vertices

The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

1

5-digit numbers are divisible by 3 and contain the digit 6

How many five digit numbers are divisible by

3

and contain the digit

6

1

compare diagonals in trapezoid

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png