2014 HMIC

Part of Harvard-MIT Mathematics Tournament

Subcontests

(5)

1

2014 HMIC #5

n

be a positive integer, and let

A

B

n\times n

matrices with complex entries such that

A^2=B^2

. Show that there exists an

n\times n

invertible matrix

S

with complex entries that satisfies

S(AB-BA)=(BA-AB)S

1

2014 HMIC #4

\omega

be a root of unity and

f

be a polynomial with integer coefficients. Show that if

|f(\omega)|=1

f(\omega)

is also a root of unity.

1

2014 HMIC #3

Fix positive integers

m

n

a_1, a_2, \dots, a_m

are reals, and that pairwise distinct vectors

v_1, \dots, v_m\in \mathbb{R}^n

\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0

i=1,2,\dots,m

\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.

1

2014 HMIC #2

2014

triangles have non-overlapping interiors contained in a circle of radius

1

. What is the largest possible value of the sum of their areas?

1

2014 HMIC #1

Consider a regular

n

n>3

, call a line acceptable if it passes through the interior of this

n

m

different acceptable lines, so that the

n

-gon is divided into several smaller polygons.
(a) Prove that there exists an

m

, depending only on

n

, such that any collection of

m

acceptable lines results in one of the smaller polygons having

3

4

sides.
(b) Find the smallest possible

m

which guarantees that at least one of the smaller polygons will have

3

4