MathDB

1

Three half-circles - find the length of AB

In the figure below, you see three half-circles. The circle

C

is tangent to two of the half-circles and to the line

PQ

perpendicular to the diameter

AB

. The area of the shaded region is

39\pi

, and the area of the circle

C

is

9\pi

. Find the length of the diameter

AB

.

19

1

Four heaps of matches two-player games

Four heaps contain

38,45,61

and

70

matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?

17

1

Making the numbers A,B,C,D from the digits 1,2,3...9 once

Using each of the eight digits

1,3,4,5,6,7,8

and

9

exactly once, a three-digit number

A

, two two-digit numbers

B

and

C

,

B<C

, and a one digit number

D

are formed. The numbers are such that

A+D=B+C=143

. In how many ways can this be done?

16

1

Filling a 2x2 square two-player game

On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses

\times

, the other

\circ

. The first who fills a

2\times 2

square with his symbols wins. Can the player who starts always win?

15

1

Find a,b for which cyclic x_i inequality holds

For which positive real numbers

a,b

does the inequality

x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a

hold for all integers

n>2

and positive real numbers

x_1,\ldots ,x_n

?

14

1

Find the cot sum

The graph of the function

f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0

(where

n>1

) intersects the line

y=b

at the points

B_1,B_2,\ldots ,B_n

(from left to right), and the line

y=c\ (c\not= b)

at the points

C_1,C_2,\ldots ,C_n

(from left to right). Let

P

be a point on the line

y=c

, to the right to the point

C_n

. Find the sum

\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP)

13

1

f(x)=f(x^2+x+1) when f is either odd or even

Consider the functions

f

defined on the set of integers such that

f(x)=f(x^2+x+1)

for all integer

x

. Find

(a)

all even functions,

(b)

all odd functions of this kind.

12

1

-2 belongs to S

Let

S

be a set of integers containing the numbers

0

and

1996

. Suppose further that any integer root of any non-zero polynomial with coefficients in

S

also belongs to

S

. Prove that

-2

belongs to

S

.

11

1

At least three of x_i are equal

Real numbers

x_1,x_2,\ldots ,x_{1996}

have the following property: For any polynomial

W

of degree

2

at least three of the numbers

W(x_1),W(x_2),\ldots ,W(x_{1996})

are equal. Prove that at least three of the numbers

x_1,x_2,\ldots ,x_{1996}

are equal.

10

1

d(a^n-1) is at least n if a^n+1 is prime

Denote by

d(n)

the number of distinct positive divisors of a positive integer

n

(including

1

and

n

). Let

a>1

and

n>0

be integers such that

a^n+1

is a prime. Prove that

d(a^n-1)\ge n

.

9

1

Find the integer b and set A satisfying the three conditions

Let

n

and

k

be integers,

1\le k\le n

. Find an integer

b

and a set

A

of

n

integers satisfying the following conditions: (i) No product of

k-1

distinct elements of

A

is divisible by

b

. (ii) Every product of

k

distinct elements of

A

is divisible by

b

. (iii) For all distinct

a,a'

in

A

,

a

does not divide

a'

.

7

1

None of the terms in the inductive sequence a_i are 0

A sequence of integers

a_1,a_2,\ldots

is such that

a_1=1,a_2=2

and for

n\ge 1

,

a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right.

Prove that

a_n\not= 0

for all

n

.

5

1

Japanese-Inspired Baltic Way problem

Let

ABCD

be a cyclic convex quadrilateral and let

r_a,r_b,r_c,r_d

be the radii of the circles inscribed in the triangles

BCD, ACD, ABD, ABC

, respectively. Prove that

r_a+r_c=r_b+r_d

.

4

1

P,Q on AB,CD maximising the (equal) angles CPD,BQA

ABCD

is a trapezium where

AD\parallel BC

.

P

is the point on the line

AB

such that

\angle CPD

is maximal.

Q

is the point on the line

CD

such that

\angle BQA

is maximal. Given that

P

lies on the segment

AB

, prove that

\angle CPD=\angle BQA

.

3

1

Find all possible values of segment PQ

Let

ABCD

be a unit square and let

P

and

Q

be points in the plane such that

Q

is the circumcentre of triangle

BPC

and

D

be the circumcentre of triangle

PQA

. Find all possible values of the length of segment

PQ

.

1

α/β is a rational number in regular 1996-gon

Let

\alpha

be the angle between two lines containing the diagonals of a regular

1996

-gon, and let

\beta\not= 0

be another such angle. Prove that

\frac{\alpha}{\beta}

is a rational number.

6

1

a+b+c+d is not prime if ab=cd

Let

a,b,c,d

be positive integers such that ab\equal{}cd. Prove that a\plus{}b\plus{}c\plus{}d is a composite number.

8

1

GCD of 1995 and 1996 term of the sequence - Baltic Way 1996

Consider the sequence:

x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n

, for

n>1

, where

\text{lcm} (a,b)

means the least common multiple of

a

and

b

. Find the greatest common divisor of

x_{1995}

and

x_{1996}

.

20

1

Paritioning positive integers into A and B with conditions

Is it possible to partition all positive integers into disjoint sets

A

and

B

such that (i) no three numbers of

A

form an arithmetic progression, (ii) no infinite non-constant arithmetic progression can be formed by numbers of

B

?

18

1

competent

The jury of an Olympiad has

30

members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most

15

sessions there will be no more exclusions. (Note that nobody votes about his own competence.)

1996 Baltic Way

Subcontests

Three half-circles - find the length of AB

Four heaps of matches two-player games

Making the numbers A,B,C,D from the digits 1,2,3...9 once

Filling a 2x2 square two-player game

Find a,b for which cyclic x_i inequality holds

Find the cot sum

f(x)=f(x^2+x+1) when f is either odd or even

-2 belongs to S

At least three of x_i are equal

d(a^n-1) is at least n if a^n+1 is prime

Find the integer b and set A satisfying the three conditions

None of the terms in the inductive sequence a_i are 0

Japanese-Inspired Baltic Way problem

P,Q on AB,CD maximising the (equal) angles CPD,BQA

Find all possible values of segment PQ

α/β is a rational number in regular 1996-gon

a+b+c+d is not prime if ab=cd

GCD of 1995 and 1996 term of the sequence - Baltic Way 1996

Paritioning positive integers into A and B with conditions

competent