2019 USAJMO

Part of USAJMO

Subcontests

(6)

1

Poset on subsets of 2n element set

n

be a nonnegative integer. Determine the number of ways that one can choose

(n+1)^2

S_{i,j}\subseteq\{1,2,\ldots,2n\}

i,j

0\leq i,j\leq n

, such that:
[*] for all

0\leq i,j\leq n

S_{i,j}

i+j

elements; and [*]

S_{i,j}\subseteq S_{k,l}

0\leq i\leq k\leq n

0\leq j\leq l\leq n

.
Proposed by Ricky Liu

1

Excircle

ABC

be a triangle with

\angle ABC

A

-excircle is a circle in the exterior of

\triangle ABC

that is tangent to side

BC

of the triangle and tangent to the extensions of the other two sides. Let

E

F

be the feet of the altitudes from

B

C

AC

AB

, respectively. Can line

EF

be tangent to the

A

-excircle?
Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal

1

Evan's mean blackboard game

Two rational numbers

\tfrac{m}{n}

\tfrac{n}{m}

are written on a blackboard, where

m

n

are relatively prime positive integers. At any point, Evan may pick two of the numbers

x

y

written on the board and write either their arithmetic mean

\tfrac{x+y}{2}

or their harmonic mean

\tfrac{2xy}{x+y}

on the board as well. Find all pairs

(m,n)

such that Evan can write 1 on the board in finitely many steps.
Proposed by Yannick Yao

1

Functional Equations

\mathbb{Z}

be the set of all integers. Find all pairs of integers

(a,b)

for which there exist functions

f \colon \mathbb{Z}\rightarrow \mathbb{Z}

g \colon \mathbb{Z} \rightarrow \mathbb{Z}

satisfying f(g(x))=x+a \text{and} g(f(x))=x+b for all integers

x

.
Proposed by Ankan Bhattacharya

1

Apples and Pears

a+b

bowls arranged in a row, numbered

1

a+b

a

b

are given positive integers. Initially, each of the first

a

bowls contains an apple, and each of the last

b

bowls contains a pear. A legal move consists of moving an apple from bowl

i

i+1

and a pear from bowl

j

j-1

, provided that the difference

i-j

is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first

b

bowls each containing a pear and the last

a

bowls each containing an apple. Show that this is possible if and only if the product

ab

1

The best length condition you'll ever see

ABCD

be a cyclic quadrilateral satisfying

AD^2 + BC^2 = AB^2

. The diagonals of

ABCD

E

P

be a point on side

\overline{AB}

\angle APD = \angle BPC

. Show that line

PE

\overline{CD}

.
Proposed by Ankan Bhattacharya