2017 USAMO

Part of USAMO

Subcontests

(4)

1

Looks Like Mount Inequality Erupted :(

Find the minimum possible value of

\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}

a

b

c

d

are nonnegative real numbers such that

a+b+c+d=4

.
Proposed by Titu Andreescu

1

DOTS EVERYWHERE

\mathbf{Z}

denote the set of all integers. Find all real numbers

c > 0

such that there exists a labeling of the lattice points

( x, y ) \in \mathbf{Z}^2

with positive integers for which:
[*] only finitely many distinct labels occur, and [*] for each label

i

, the distance between any two points labeled

i

c^i

.
Proposed by Ricky Liu

1

Inversions of Permutations

m_1, m_2, \ldots, m_n

be a collection of

n

positive integers, not necessarily distinct. For any sequence of integers

A = (a_1, \ldots, a_n)

and any permutation

w = w_1, \ldots, w_n

m_1, \ldots, m_n

A

w

to be a pair of entries

w_i, w_j

i < j

for which one of the following conditions holds:
[*]

a_i \ge w_i > w_j

w_j > a_i \ge w_i

w_i > w_j > a_i

.
Show that, for any two sequences of integers

A = (a_1, \ldots, a_n)

B = (b_1, \ldots, b_n)

, and for any positive integer

k

, the number of permutations of

m_1, \ldots, m_n

k

A

-inversions is equal to the number of permutations of

m_1, \ldots, m_n

k

B

1

MAN IS KID

ABC

be a scalene triangle with circumcircle

\Omega

I

AI

\overline{BC}

D

\Omega

M

; the circle with diameter

\overline{DM}

\Omega

K

MK

BC

S

N

is the midpoint of

\overline{IS}

. The circumcircles of

\triangle KID

\triangle MAN

intersect at points

L_1

L_2

\Omega

passes through the midpoint of either

\overline{IL_1}

\overline{IL_2}

.
Proposed by Evan Chen