2017 Lusophon Mathematical Olympiad

Part of Lusophon Mathematical Olympiad

Subcontests

(6)

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VII Lusophon Mathematical Olympiad 2017 - Problem 4

Find how many multiples of 360 are of the form

\overline{ab2017cd}

, where a, b, c, d are digits, with a > 0.

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VII Lusophon Mathematical Olympiad 2017 - Problem 6

Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that

\overline{AF}

\overline{DF}

\overline{BE}

\overline{DE}

. Show that the circumcenter of ABC lies on the circumcircle of CEF.

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VII Lusophon Mathematical Olympiad 2017 - Problem 5

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.

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VII Lusophon Mathematical Olympiad 2017 - Problem 1

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.

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VII Lusophon Mathematical Olympiad 2017 - Problem 2

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.

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VII Lusophon Mathematical Olympiad 2017 - Problem 3

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.