MathDB

1

Expected number of prize winners after dealing cards twice

N

cards, the cards are denoted by

1

to

N

. These cards are dealt to

N

people twice. A person

X

wins a prize if there is no person

Y

who got a card with a smaller number than

X

both times. Determine the expected number of prize winners.

3

1

Defining the points P_i in triangle ABC so P_6=P_1

Let

P_0

be a point in the plane of triangle

A_0A_1A_2

. Define

P_i\ (i=1,\ldots ,6)

inductively as the point symmetric to

P_{i-1}

with respect to

A_k

, where

k

is the remainder when

i

is divided by

3

.
a) Prove that

P_6\equiv P_1

. b) Find the locus of points

P_0

for which

P_iP_{i+1}

does not meet the interior of

\triangle A_0A_1A_2

for

0\le i\le 5

.

2

1

If l_{i}^2 are all rational then so is l_{10}^2

Five points, no three collinear, are given on the plane. Let

l_1,l_2,\ldots ,l_{10}

be the lengths of the ten segments joining any two of the given points. Prove that if

l_1^2,\ldots ,l_9^2

are rational numbers, then

l_{10}^2

is also a rational number.

1

Sequence (c_n) that does not cross with sequence (b_n)

For any positive integer

n

, let

a_n

denote the closest integer to

\sqrt{n}

, and let

b_n=n+a_n

. Determine the increasing sequence

(c_n)

of positive integers which do not occur in the sequence

(b_n)

.

4

1

Geometry

In a triangle

ABC

,

M

is the midpoint of

BC

. Given that

\angle MAC=15^{\circ}

, find the maximum possible value of

\angle ABC

.

1994 Japan MO Finals

Subcontests

Expected number of prize winners after dealing cards twice

Defining the points P_i in triangle ABC so P_6=P_1

If l_{i}^2 are all rational then so is l_{10}^2

Sequence (c_n) that does not cross with sequence (b_n)

Geometry