Find all positive integers such that

Let , , , be four points such that no three are collinear and is not the orthocenter of . Let , , be the orthocenters of , , , respectively. Suppose that the lines , , are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.

Andrew Gu

Let and be the circumcenter and orthocenter, respectively, of an acute scalene triangle . The perpendicular bisector of intersects and at and respectively. Let denote the intersection of the circumcircles of triangles and other than .

Define and analogously by repeating this construction two more times. Prove that , , , and are concyclic.

Hongzhou Lin

Let a_k denote the first non zero digit of the decimal representation of k!. Does the sequence a_1, a_2, a_3, … eventually become periodic?

Let be a quadrilateral inscribed in a circle . Let , , , , , , , . Prove that lies over .

For an table filled with natural numbers, we say it is a divisor table if:
- the numbers in the -th row are exactly all the divisors of some natural number ,
- the numbers in the -th column are exactly all the divisors of some natural number ,
- for every .

A prime number is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.

Proposed by Pavle Martinović

Let be an odd prime and be integers so that the integers are divisible by .
Prove that divides each of .

Proposed by Navilarekallu Tejaswi

Let be a fixed integer. There are beads on a circular necklace. You wish to paint the beads using colors, such that among any consecutive beads every color appears at least once. Find the largest value of for which this task is possible.

Carl Schildkraut, USA

Let be a triangle (). are the altitudes of the triangle. The bisector of intersects at . Let be a point such that and . Prove that passes through the midpoint of .

Proposed by Iman Maghsoudi, Hooman Fattahi

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