2014 USAMO

Part of USAMO

Subcontests

(4)

1

Segment has Length Equal to Circumradius

ABC

be a triangle with orthocenter

H

P

be the second intersection of the circumcircle of triangle

AHC

with the internal bisector of the angle

\angle BAC

X

be the circumcenter of triangle

APB

Y

the orthocenter of triangle

APC

. Prove that the length of segment

XY

is equal to the circumradius of triangle

ABC

1

nth root of a, b in O(sqrt(n))

Prove that there is a constant

c>0

with the following property: If

a, b, n

are positive integers such that

\gcd(a+i, b+j)>1

i, j\in\{0, 1, \ldots n\}

\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.

1

Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points

\dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots

in the plane with the following property: For any three distinct integers

a,b,

c

P_a

P_b

P_c

are collinear if and only if

a+b+c=2014

1

Inequality on a Quartic

a

b

c

d

be real numbers such that

b-d \ge 5

x_1, x_2, x_3,

x_4

of the polynomial

P(x)=x^4+ax^3+bx^2+cx+d

are real. Find the smallest value the product

(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)