2015 USAJMO

Part of USAJMO

Subcontests

(6)

1

Points with equal angles

ABCD

be a cyclic quadrilateral. Prove that there exists a point

X

\overline{BD}

\angle BAC=\angle XAD

\angle BCA=\angle XCD

if and only if there exists a point

Y

\overline{AC}

\angle CBD=\angle YBA

\angle CDB=\angle YDA

1

Rocks and Squares

Steve is piling

m\geq 1

indistinguishable stones on the squares of an

n\times n

grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions

(i, k), (i, l), (j, k), (j, l)

1\leq i, j, k, l\leq n

i<j

k<l

. A stone move consists of either removing one stone from each of

(i, k)

(j, l)

and moving them to

(i, l)

(j, k)

respectively, or removing one stone from each of

(i, l)

(j, k)

and moving them to

(i, k)

(j, l)

respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?

1

Sort of additive function

Find all functions

f:\mathbb{Q}\rightarrow\mathbb{Q}

f(x)+f(t)=f(y)+f(z)

for all rational numbers

x<y<z<t

that form an arithmetic progression. (

\mathbb{Q}

is the set of all rational numbers.)

1

Sequences of real numbers

Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence -- no matter what move -- there is always a way to continue with a finite sequence of moves so as to obtain in the end a constant sequence.

1

Quadrilateral APBQ

APBQ

is inscribed in circle

\omega

\angle P = \angle Q = 90^{\circ}

AP = AQ < BP

X

be a variable point on segment

\overline{PQ}

AX

\omega

S

A

T

AQB

\omega

\overline{XT}

is perpendicular to

\overline{AX}

M

denote the midpoint of chord

\overline{ST}

X

varies on segment

\overline{PQ}

M

moves along a circle.

1

Another Cubic Curve!

Solve in integers the equation

x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.