5

Part of ICMC 7

Problems(2)

Representing integers using integer polynomials

Source: ICMC 7 Round 1 Problem 5

[*]Is there a non-linear integer-coefficient polynomial

P(x)

N{}

such that all integers greater than

N{}

may be written as the greatest common divisor of

P(a){}

P(b){}

for positive integers

a>b

? [*]Is there a non-linear integer-coefficient polynomial

Q(x)

M{}

such that all integers greater than

M{}

may be written as

Q(a) - Q(b)

for positive integers

a>b

? Proposed by Dylan Toh

number theorypolynomialICMC

Equilateral triangle dissection

Source: ICMC 7 Round 2 Problem 5

Is it possible to dissect an equilateral triangle into three congruent polygonal pieces (not necessarily convex), one of which contains the triangle’s centre in its interior?
Note: The interior of a polygon is the polygon without its boundary.
Proposed by Dylan Toh