MathDB

2014 Algebra #8: True for Exactly One Value

Source:

7/7/2014

Find all real numbers

k

such that

r^4+kr^3+r^2+4kr+16=0

is true for exactly one real number

r

.

symmetryalgebrapolynomialinequalities

2014 Combinatorics #8: Chessboard Diagonal Sum

Source:

2/23/2014

The integers

1, 2, \dots, 64

are written in the squares of a

8 \times 8

chess board, such that for each

1 \le i < 64

, the numbers

i

and

i+1

are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?

2014 Geometry #8: Tangent to the Circumcircle

Source:

2/25/2014

Let

ABC

be a triangle with sides

AB = 6

,

BC = 10

, and

CA = 8

. Let

M

and

N

be the midpoints of

BA

and

BC

, respectively. Choose the point

Y

on ray

CM

so that the circumcircle of triangle

AMY

is tangent to

AN

. Find the area of triangle

NAY

.

geometrycircumcircletrigonometryanalytic geometrygraphing lines

2014 Guts #8: Numbers on Blackboard

Source:

2/25/2014

The numbers

2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536

are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?

2014 Team #8: Parallels, Perpendiculars with Circumcenter

Source:

3/2/2014

Let

ABC

be an acute triangle with circumcenter

O

such that

AB=4

,

AC=5

, and

BC=6

. Let

D

be the foot of the altitude from

A

to

BC

, and

E

be the intersection of

AO

with

BC

. Suppose that

X

is on

BC

between

D

and

E

such that there is a point

Y

on

AD

satisfying

XY\parallel AO

and

YO\perp AX

. Determine the length of

BX

.

geometrycircumcirclesymmedianAngle Chasingradical axis

8

Problems(5)