2017 JBMO Problems
Problem 1
Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers.
Problem 2
Let be positive integers such that .Prove that When does the equality hold?
Problem 3
Let be an acute triangle such that ,with circumcircle and circumcenter . Let be the midpoint of and be a point on such that . let be a point such that is a parallelogram and a point on the same side of as such that and . Let the line intersect at and let the circumcircle of intersect at point . Prove that the point and are collinear .
Problem 4
Consider a regular 2n-gon , in the plane ,where is a positive integer . We say that a point on one of the sides of can be seen from a point that is external to , if the line segment contains no other points that lie on the sides of except .We color the sides of in 3 different colors (ignore the vertices of ,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to , points of most 2 different colors on can be seen .Find the number of distinct such colorings of (two colorings are considered distinct if at least one of sides is colored differently).
See also
2017 JBMO (Problems • Resources) | ||
Preceded by 2016 JBMO Problems |
Followed by 2018 JBMO Problems | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |