2003 El Salvador Correspondence / Qualifying NMO III
Source:
October 15, 2021
algebrageometrycombinatoricsnumber theoryel salvador NMO
Problem Statement
p1. Determine the smallest natural number that has the property that it's cube ends in .
p2. Triangle is isosceles with . The angle bisector at intercepts side at the point and the bisector of the angle at intercepts side at . If , find the measure of the angles of triangle .
p3.In the accompanying figure, the circle is tangent to the quadrant at point . AP is tangent to at and is tangent at . Calculate the length as a function of the radius of the quadrant .
https://cdn.artofproblemsolving.com/attachments/3/c/d34774c3c6c33d351316574ca3f7ade54e6441.png
p4. Let be a set with seven or more natural numbers. Show that there must be two numbers in with the property that either its sum or its difference is divisible by ten. Show that the property can fail if set has fewer than seven elements.
p5. An -sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals.