OMCPLP 2017

In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points. If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points. Instead, if we exchange the questions scores, scoring D points for the easy questions and 3 for the hard ones, the total punctuation of the test is multiplied by $\frac{3}{2}$ . Knowing that the number of easy questions is 9 times bigger the number of hard questions, find the number of questions in this test.

Let ABCD be a parallelogram, E the midpoint of AD and F the projection of B on CE. Prove that the triangle ABF is isosceles.

Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.

Find how many multiples of 360 are of the form $\overline{ab2017cd}$ , where a, b, c, d are digits, with a > 0.

The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.

Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$ = $\overline{DF}$ and $\overline{BE}$ = $\overline{DE}$ . Show that the circumcenter of ABC lies on the circumcircle of CEF.