MathDB

1

Part of 1987 National High School Mathematics League

Problems(2)

Perfect Cube

Source: 1987 National High School Mathematics League, Exam One, Problem 1

2/24/2020
For any given positive integer nn, n6+3an^6+3a is a perfect cube, where aa is a positive integer. Then (A)\text{(A)}There is no such aa. (B)\text{(B)}There are infinitely many such aa. (C)\text{(C)}There is finitely many such aa. (D)\text{(D)}None of (A)(B)(C)\text{(A)(B)(C)} is correct.
Geometry Problem

Source: 1987 National High School Mathematics League, Exam Two, Problem 1

2/24/2020
△ABC\triangle ABC and △ADE\triangle ADE (∠ABC=∠ADE=π2)(\angle ABC=\angle ADE=\frac{\pi}{2}) are two isosceles right triangle that are not congruent. Fix △ABC\triangle ABC, but rotate △ADE\triangle ADE on the plane. Prove that there exists point M∈BCM\in BC, satisfying that △BMD\triangle BMD is an isosceles right triangle.
geometryrotation