MathDB

2014 Algebra #7: Inequality involving Median

Source:

7/7/2014

Find the largest real number

c

such that

\sum_{i=1}^{101}x_i^2\geq cM^2

whenever

x_1,\ldots,x_{101}

are real numbers such that

x_1+\cdots+x_{101}=0

and

M

is the median of

x_1,\ldots,x_{101}

.

inequalities

2014 Combinatorics #7: Tournament Outcomes

Source:

2/23/2014

Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever

A

and

B

are players in the tournament for which

A

won (strictly) more matches than

B

over the course of the tournament, it is also the case that

A

won the match against

B

during the tournament. In how many ways could the tournament have gone?

countingdistinguishability

2014 Geometry #7: 13-14-15 Circle

Source:

2/25/2014

Triangle

ABC

has sides

AB = 14

,

BC = 13

, and

CA = 15

. It is inscribed in circle

\Gamma

, which has center

O

. Let

M

be the midpoint of

AB

, let

B'

be the point on

\Gamma

diametrically opposite

B

, and let

X

be the intersection of

AO

and

MB'

. Find the length of

AX

.

geometryanalytic geometrycircumcircleperpendicular bisector

2014 Guts #7: Evil League of Evil

Source:

2/25/2014

The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at

(5, 1)

and put poison in two pipes, one along the line

y=x

and one along the line

x=7

. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?

geometrygeometric transformationreflection

2014 Team #7: Maximum Number of Equal Diagonals

Source:

3/2/2014

Find the maximum possible number of diagonals of equal length in a convex hexagon.

7

Problems(5)