2013 USAJMO

Part of USAJMO

Subcontests

(5)

1

Square Root Equality

Find all real numbers

x,y,z\geq 1

\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.

1

Quadrilateral Inscribed in Semicircle

XABY

is inscribed in the semicircle

\omega

XY

AY

BX

P

Z

is the foot of the perpendicular from

P

XY

C

\omega

XC

is perpendicular to line

AZ

Q

be the intersection of segments

AY

XC

\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.

1

Partitions into Powers of 2

f(n)

be the number of ways to write

n

as a sum of powers of

2

, where we keep track of the order of the summation. For example,

f(4)=6

4

can be written as

4

2+2

2+1+1

1+2+1

1+1+2

1+1+1+1

. Find the smallest

n

2013

f(n)

1

The Miquel Point "Returns"

ABC

P

Q

R

BC

CA

AB

respectively. Let

\omega_A

\omega_B

\omega_C

denote the circumcircles of triangles

AQR

BRP

CPQ

, respectively. Given the fact that segment

AP

\omega_A

\omega_B

\omega_C

X

Y

Z

, respectively, prove that

YX/XZ=BP/PC

1

Gardens of Rectangular Grids

Each cell of an

m\times n

board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either

0

1

. (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to

0

.
Determine the number of distinct gardens in terms of

m

n