2020 USOJMO

Part of USAJMO

Subcontests

(6)

1

Factoring Multivariable Polynomials

n \geq 2

be an integer. Let

P(x_1, x_2, \ldots, x_n)

be a nonconstant

n

-variable polynomial with real coefficients. Assume that whenever

r_1, r_2, \ldots , r_n

are real numbers, at least two of which are equal, we have

P(r_1, r_2, \ldots , r_n) = 0

P(x_1, x_2, \ldots, x_n)

cannot be written as the sum of fewer than

n!

monomials. (A monomial is a polynomial of the form

cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n

c

is a nonzero real number and

d_1

d_2

\ldots

d_n

are nonnegative integers.)
Proposed by Ankan Bhattacharya

1

geo equals ForeBoding For Dennis

ABCD

be a convex quadrilateral inscribed in a circle and satisfying

DA < AB = BC < CD

E

F

are chosen on sides

CD

AB

BE \perp AC

EF \parallel BC

FB = FD

1

apparently circles have two intersections :'(

\omega

be the incircle of a fixed equilateral triangle

ABC

\ell

be a variable line that is tangent to

\omega

and meets the interior of segments

BC

CA

P

Q

, respectively. A point

R

is chosen such that

PR = PA

QR = QB

. Find all possible locations of the point

R

, over all choices of

\ell

.
Proposed by Titu Andreescu and Waldemar Pompe

1

NT, Combo, or Geo?

(a_1,b_1),

(a_2,b_2),

\dots,

(a_{100},b_{100})

are distinct ordered pairs of nonnegative integers. Let

N

denote the number of pairs of integers

(i,j)

1\leq i<j\leq 100

|a_ib_j-a_jb_i|=1

. Determine the largest possible value of

N

over all possible choices of the

100

ordered pairs.
Proposed by Ankan Bhattacharya

1

A VERY Long Bookshelf

n \geq 2

be an integer. Carl has

n

books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right.
Proposed by Milan Haiman

1

Beams inside a cube

2020 \times 2020 \times 2020

cube is given, and a

2020 \times 2020

grid of square unit cells is drawn on each of its six faces. A beam is a

1 \times 1 \times 2020

rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two

1 \times 1

faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are

3 \cdot {2020}^2

possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four

1 \times 2020

faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Proposed by Alex Zhai