1998 Chile Classification / Qualifying NMO X
Source:
October 8, 2021
algebrageometrycombinatoricsnumber theorychilean NMO
Problem Statement
p1. From each natural number less than , two digits are subtracted from the sum of the squares. For what values of is this difference maximum?
p2. In a triangle , isosceles and right in , a point is taken on the hypotenuse and is projected perpendicularly at points and of legs and , respectively, dividing thus the triangle in the triangles , and the rectangle . If each leg measures , show that at least one of the areas of these three figures of this division is greater than or equal to .
p3. A normal year has days. A leap year has days, a leap year being a year not divisible by , except those that being divisible by are not divisible by (for example 1900 was not leap, but the year will be such). Remembering that today is Saturday August , , what day of the week was September , ?
p4. Miriam invited nine boys and eight girls for her birthday. Her mother prepared T-shirts with the numbers from to and distributed them to all the participants of this. During a dance, the mother observed that the sum of the numbers of each pair was a perfect square. How were the nine couples made up?
p5. Let be the point where the altitudes of a triangle intersect. Show that the angle formed by the radii of the circumferences of diameter and at their points of intersection, is a right angle.
p6.In how many ways can people be photographed, sitting in a row, so that four of them and always remain in the same relative order, that is, is always at the left of , that is always to the left of and that is always to the left of ?
p7. Find a number that is divisible by and such that the sum of its digits when written in the decimal system it is equal to .