2019 Ecuador NMO (OMEC)

Part of Ecuador Mathematical Olympiad (OMEC)

Subcontests

(4)

1

rational painted area, circles, regular polygon 2019 Ecuador NMO (OMEC) 3.6

n\ge 3

be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of

n

vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the

n

small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.

1

a + b + c=? if 4abc=(a+3) (b+3)(c+3) 2018 Ecuador NMO (OMEC) 3.5

a, b, c

be integers not all the same with

a, b, c\ge 4

4abc = (a + 3) (b + 3) (c + 3).

Find the numerical value of

a + b + c

1

guess the sum of digits game 2019 Ecuador NMO (OMEC) 3.4

n> 1

be a positive integer. Danielle chooses a number

N

n

digits but does not tell her students and they must find the sum of the digits of

N

. To achieve this, each student chooses and says once a number of

n

digits to Danielle and she tells how many digits are in the correct location compared with

N

. Find the minimum number of students that must be in the class to ensure that students have a strategy to correctly find the sum of the digits of

N

in any case and show a strategy in that case.

1

max 2^k divides 1+2019+...+2019^{n-1} --- 2019 Ecuador NMO (OMEC) 3.3

For every positive integer

n

, find the maximum power of

2

that divides the number

1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.