4

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #4: Homogeneous polynomials

Source:

P(x, y, z)

is a homogeneous degree 4 polynomial in three variables such that

P(a, b, c) = P(b, c, a)

P(a, a, b) = 0

a

b

c

P(1, 2, 3) = 1

P(2, 4, 8)

P(x, y, z)

is a homogeneous degree

4

polynomial if it satisfies

P(ka, kb, kc) = k^4P(a, b, c)

k, a, b, c

polynomialalgebra

2021 Combo #4: Counting Functions

Source:

S = \{1, 2, \dots, 9\}.

Compute the number of functions

f : S \rightarrow S

such that, for all

s \in S, f(f(f(s))) =s

f(s) - s

is not divisible by

3

2021 Geo #4: Trapezoidzzzz

Source:

Let ABCD be a trapezoid with

AB \parallel CD, AB = 5, BC = 9, CD = 10,

DA = 7

BC

DA

intersect at point

E

M

be the midpoint of

CD

N

be the intersection of the circumcircles of

\triangle BMC

\triangle DMA

M

EN^2 = \tfrac ab

for relatively prime positive integers

a

b

100a + b

Source:

k

n

be positive integers and let

S=\{(a_1,\ldots,a_k)\in \mathbb{Z}^{k}\;|\; 0\leq a_k\leq\cdots\leq a_1 \leq n,a_1+\cdots+a_k=n\}

. Determine, with proof, the value of

\sum_{(a_1,\ldots,a_k)\in S}\binom{n}{a_1}\binom{a_1}{a_2}\cdots\binom{a_{k-1}}{a_k}

k

n

, where the sum is over all

k

S

Summationcombinatorics