MathDB

1

3 divides no of lists of jointly coprime positive integer numbers that sum to n

n \ge 3

, show that the number of lists of jointly coprime positive integer numbers that sum to

n

is divisible by

3

. (For instance, if

n = 4

, there are six such lists:

(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)

and

(1, 1, 1, 1)

.)

3

1

exists f:R \to R, f(x)+f(2x)+...+f(nx) = 0 for all x \in R, f(x) = 0 iff x=0

Given an integer number

n \ge 2

, show that there exists a function

f : R \to R

such that

f(x) + f(2x) + ...+ f(nx) = 0

, for all

x \in R

, and

f(x) = 0

if and only if

x = 0

.

2

1

area <=4, non-zero lattice points lie outside the polygon, 4 quadrants

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than

4

.

1

locus of concurrency point as a line turns about a point

Let

A_0A_1A_2

be a triangle and let

P

be a point in the plane, not situated on the circle

A_0A_1A_2

. The line

PA_k

meets again the circle

A_0A_1A_2

at point

B_k, k = 0, 1, 2

. A line

\ell

through the point

P

meets the line

A_{k+1}A_{k+2}

at point

C_k, k = 0, 1, 2

. Show that the lines

B_kC_k, k = 0, 1, 2

, are concurrent and determine the locus of their concurrency point as the line

\ell

turns about the point

P

.

2011 IMAR Test

Subcontests

3 divides no of lists of jointly coprime positive integer numbers that sum to n

exists f:R \to R, f(x)+f(2x)+...+f(nx) = 0 for all x \in R, f(x) = 0 iff x=0

area <=4, non-zero lattice points lie outside the polygon, 4 quadrants

locus of concurrency point as a line turns about a point

2011 IMAR Test

Subcontests

3 divides no of lists of jointly coprime positive integer numbers that sum to n

exists f:R \to R, f(x)+f(2x)+...+f(nx) = 0 for all x \in R, f(x) = 0 iff x=0

area &lt;=4, non-zero lattice points lie outside the polygon, 4 quadrants

locus of concurrency point as a line turns about a point

area <=4, non-zero lattice points lie outside the polygon, 4 quadrants