MathDB

Frog jumping on points on a plane

Source: India Postals 2015 Set 1

11/7/2015

For each point

X

in the plane, a real number

r_X > 0

is assigned such that

2|r_X - r_Y | \le |XY |

, for any two points

X, Y

. (Here

|XY |

denotes the distance between

X

and

Y

) A frog can jump from

X

to

Y

if

r_X = |XY |

. Show that for any two points

X

and

Y

, the frog can jump from

X

to

Y

in a finite number of steps.

combinatorics

Hard geometry

Source: Komal,Jan 2015

5/16/2015

Let

ABCD

be a convex quadrilateral. In the triangle

ABC

let

I

and

J

be the incenter and the excenter opposite to vertex

A

, respectively. In the triangle

ACD

let

K

and

L

be the incenter and the excenter opposite to vertex

A

, respectively. Show that the lines

IL

and

JK

, and the bisector of the angle

BCD

are concurrent.

geometryincenter

Partitionaing a Square

Source: India Postals 2015 Set 3

11/7/2015

Suppose a

m \times m

square can be divided into

7

rectangles such that no two rectangles have a common interior point and the side-lengths of the rectangles form the set

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}

. Find the maximum value of

m

.

combinatorics

Cardinality of set of lattice points with different pairwise

Source: India Postals 2015

11/15/2015

Let

S

be a set of in

3-

space such that each of the points in

S

has integer coordinates

(x,y,z)

with

1 \le x,y,z \le n

. Suppose the pairwise distances between these points are all distinct. Prove that

|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}

analytic geometrycombinatorics

Problem 5

Problems(4)