MathDB

1

ASU 486 All Soviet Union MO 1988 r < ab/( 2(a + b) ) inradius of tetrahedron

Prove that for any tetrahedron the radius of the inscribed sphere

r <\frac{ ab}{ 2(a + b)}

, where

a

and

b

are the lengths of any pair of opposite edges.

485

1

ASU 485 All Soviet Union MO 1988 a_0 = 0, a_n = p(a_n-1), p(x) pos.integer pol

The sequence of integers an is given by

a_0 = 0, a_n = p(a_n-1)

, where

p(x)

is a polynomial whose coefficients are all positive integers. Show that for any two positive integers

m, k

with greatest common divisor

d

, the greatest common divisor of

a_m

and

a_k

is

a_d

.

484

1

ASU 484 All Soviet Union MO 1988 sinx_1+2sin x_2+ ...+nsinx_n=100

What is the smallest

n

for which there is a solution to

\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}

?

483

1

ASU 483 All Soviet Union MO 1988 broken line with vertices on parabola

A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.

482

1

ASU 482 All Soviet Union MO 1988 1+2+... +n = mk, k groups division

Let

m, n, k

be positive integers with

m \ge n

and

1 + 2 + ... + n = mk

. Prove that the numbers

1, 2, ... , n

can be divided into

k

groups in such a way that the sum of the numbers in each group equals

m

.

481

1

ASU 481 All Soviet Union MO 1988 broken line between opposite vertices of cube

A polygonal line connects two opposite vertices of a cube with side

2

. Each segment of the line has length

3

and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

480

1

ASU 480 All Soviet Union MO 1988 min of xy/z+yz/x+zx/y if x^2+y^2+z^2=1

Find the minimum value of

\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}

for positive reals

x, y, z

with

x^2 + y^2 + z^2 = 1

.

479

1

ASU 479 All Soviet Union MO 1988 prove that EF = DG, altitudes are equal

In the acute-angled triangle

ABC

, the altitudes

BD

and

CE

are drawn. Let

F

and

G

be the points of the line

ED

such that

BF

and

CG

are perpendicular to

ED

. Prove that

EF = DG

.

478

1

ASU 478 All Soviet Union MO 1988 n^2 reals in a nxn square table

n^2

real numbers are written in a square

n \times n

table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are

a_{ij}

and we select row

i

, column

h

and column

k

, then column h becomes

a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}

, column

k

becomes

a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}

, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.

477

1

ASU 477 All Soviet Union MO 1988 min of b/(c + d)+c/(a + b) if b+c>= a + d

What is the minimal value of

\frac{b}{c + d} + \frac{c}{a + b}

for positive real numbers

b

and

c

and non-negative real numbers

a

and

d

such that

b + c\ge a + d

?

476

1

ASU 476 All Soviet Union MO 1988 BP bisects the angle MPN

ABC

is an acute-angled triangle. The tangents to the circumcircle at

A

and

C

meet the tangent at

B

at

M

and

N

. The altitude from

B

meets

AC

at

P

. Show that

BP

bisects the angle

MPN

475

1

ASU 475 All Soviet Union MO 1988 composites in 1^1, 1^1+2^2, 1^1+2^2+3^3, ...

Show that there are infinitely many odd composite numbers in the sequence

1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...

.

474

1

ASU 474 All Soviet Union MO 1988 find M which minimises MP

In the triangle

ABC

,

\angle C

is obtuse and

D

is a fixed point on the side

BC

, different from

B

and

C

. For any point

M

on the side

BC

, different from

D

, the ray

AM

intersects the circumcircle

S

of

ABC

at

N

. The circle through

M, D

and

N

meets

S

again at

P

, different from

N

. Find the location of the point

M

which minimises

MP

.

473

1

ASU 473 All Soviet Union MO 1988 forms 10A, 10B have 29, 32 students resp.

Form

10A

has

29

students who are listed in order on its duty roster. Form

10B

has

32

students who are listed in order on its duty roster. Every day two students are on duty, one from form

10A

and one from form

10B

. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from

10A

and one from

10B

) shared duty exactly once?

472

1

ASU 472 All Soviet Union MO 1988 2(sin A)/A +... <= (1/B + 1/C) sin A +...

A, B, C

are the angles of a triangle. Show that

2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C

471

1

ASU 471 All Soviet Union MO 1988 (1 + 1/n)^{n+1} = (1 + 1/1998)^{1998}

Find all positive integers

n

satisfying

\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}

.

470

1

ASU 470 All Soviet Union MO 1988 21 towns airlines, flights,

There are

21

towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?

469

1

ASU 469 All Soviet Union MO 1988 x^5+y^5=2x^2y^2 =>(1-xy) rational's square

If rationals

x, y

satisfy

x^5 + y^5 = 2 x^2 y^2

, show that

1-x y

is the square of a rational.

468

1

ASU 468 All Soviet Union MO 1988 blackboard game (m and n)=> m+n+mn

The numbers

1

and

2

are written on an empty blackboard. Whenever the numbers

m

and

n

appear on the blackboard the number

m + n + mn

may be written. Can we obtain :
(1)

13121

,
(2)

12131

?

467

1

ASU 467 All Soviet Union MO 1988 fixed ratio of distances of midpoints

The quadrilateral

ABCD

is inscribed in a fixed circle. It has

AB

parallel to

CD

and the length

AC

is fixed, but it is otherwise allowed to vary. If

h

is the distance between the midpoints of

AC

and

BD

and

k

is the distance between the midpoints of

AB

and

CD

, show that the ratio

h/k

remains constant.

466

1

ASU 466 All Soviet Union MO 1988 19 pos. <=88, 88 pos. <=19, same sum

Given a sequence of

19

positive integers not exceeding

88

and another sequence of

88

positive integers not exceeding

19

. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.

465

1

ASU 465 All Soviet Union MO 1988 each divides product other 2, a+b=c+1

Show that there are infinitely many triples of distinct positive integers

a, b, c

such that each divides the product of the other two and

a + b = c + 1

.

464

1

ASU 464 All Soviet Union MO 1988 quadrilaterals equal areas, area bisector

ABCD

is a convex quadrilateral. The midpoints of the diagonals and the midpoints of

AB

and

CD

form another convex quadrilateral

Q

. The midpoints of the diagonals and the midpoints of

BC

and

CA

form a third convex quadrilateral

Q'

. The areas of

Q

and

Q'

are equal. Show that either

AC

or

BD

divides

ABCD

into two parts of equal area.

463

1

ASU 463 All Soviet Union MO 1988 a book with 30 stories

A book contains

30

stories. Each story has a different number of pages under

31

. The first story starts on page

1

and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?

1988 All Soviet Union Mathematical Olympiad

Subcontests

ASU 486 All Soviet Union MO 1988 r < ab/( 2(a + b) ) inradius of tetrahedron

ASU 485 All Soviet Union MO 1988 a_0 = 0, a_n = p(a_n-1), p(x) pos.integer pol

ASU 484 All Soviet Union MO 1988 sinx_1+2sin x_2+ ...+nsinx_n=100

ASU 483 All Soviet Union MO 1988 broken line with vertices on parabola

ASU 482 All Soviet Union MO 1988 1+2+... +n = mk, k groups division

ASU 481 All Soviet Union MO 1988 broken line between opposite vertices of cube

ASU 480 All Soviet Union MO 1988 min of xy/z+yz/x+zx/y if x^2+y^2+z^2=1

ASU 479 All Soviet Union MO 1988 prove that EF = DG, altitudes are equal

ASU 478 All Soviet Union MO 1988 n^2 reals in a nxn square table

ASU 477 All Soviet Union MO 1988 min of b/(c + d)+c/(a + b) if b+c>= a + d

ASU 476 All Soviet Union MO 1988 BP bisects the angle MPN

ASU 475 All Soviet Union MO 1988 composites in 1^1, 1^1+2^2, 1^1+2^2+3^3, ...

ASU 474 All Soviet Union MO 1988 find M which minimises MP

ASU 473 All Soviet Union MO 1988 forms 10A, 10B have 29, 32 students resp.

ASU 472 All Soviet Union MO 1988 2(sin A)/A +... <= (1/B + 1/C) sin A +...

ASU 471 All Soviet Union MO 1988 (1 + 1/n)^{n+1} = (1 + 1/1998)^{1998}

ASU 470 All Soviet Union MO 1988 21 towns airlines, flights,

ASU 469 All Soviet Union MO 1988 x^5+y^5=2x^2y^2 =>(1-xy) rational's square

ASU 468 All Soviet Union MO 1988 blackboard game (m and n)=> m+n+mn

ASU 467 All Soviet Union MO 1988 fixed ratio of distances of midpoints

ASU 466 All Soviet Union MO 1988 19 pos. <=88, 88 pos. <=19, same sum

ASU 465 All Soviet Union MO 1988 each divides product other 2, a+b=c+1

ASU 464 All Soviet Union MO 1988 quadrilaterals equal areas, area bisector

ASU 463 All Soviet Union MO 1988 a book with 30 stories

1988 All Soviet Union Mathematical Olympiad

Subcontests

ASU 486 All Soviet Union MO 1988 r &lt; ab/( 2(a + b) ) inradius of tetrahedron

ASU 485 All Soviet Union MO 1988 a_0 = 0, a_n = p(a_n-1), p(x) pos.integer pol

ASU 484 All Soviet Union MO 1988 sinx_1+2sin x_2+ ...+nsinx_n=100

ASU 483 All Soviet Union MO 1988 broken line with vertices on parabola

ASU 482 All Soviet Union MO 1988 1+2+... +n = mk, k groups division

ASU 481 All Soviet Union MO 1988 broken line between opposite vertices of cube

ASU 480 All Soviet Union MO 1988 min of xy/z+yz/x+zx/y if x^2+y^2+z^2=1

ASU 479 All Soviet Union MO 1988 prove that EF = DG, altitudes are equal

ASU 478 All Soviet Union MO 1988 n^2 reals in a nxn square table

ASU 477 All Soviet Union MO 1988 min of b/(c + d)+c/(a + b) if b+c&gt;= a + d

ASU 476 All Soviet Union MO 1988 BP bisects the angle MPN

ASU 475 All Soviet Union MO 1988 composites in 1^1, 1^1+2^2, 1^1+2^2+3^3, ...

ASU 474 All Soviet Union MO 1988 find M which minimises MP

ASU 473 All Soviet Union MO 1988 forms 10A, 10B have 29, 32 students resp.

ASU 472 All Soviet Union MO 1988 2(sin A)/A +... &lt;= (1/B + 1/C) sin A +...

ASU 471 All Soviet Union MO 1988 (1 + 1/n)^{n+1} = (1 + 1/1998)^{1998}

ASU 470 All Soviet Union MO 1988 21 towns airlines, flights,

ASU 469 All Soviet Union MO 1988 x^5+y^5=2x^2y^2 =&gt;(1-xy) rational's square

ASU 468 All Soviet Union MO 1988 blackboard game (m and n)=&gt; m+n+mn

ASU 467 All Soviet Union MO 1988 fixed ratio of distances of midpoints

ASU 466 All Soviet Union MO 1988 19 pos. &lt;=88, 88 pos. &lt;=19, same sum

ASU 465 All Soviet Union MO 1988 each divides product other 2, a+b=c+1

ASU 464 All Soviet Union MO 1988 quadrilaterals equal areas, area bisector

ASU 463 All Soviet Union MO 1988 a book with 30 stories

ASU 486 All Soviet Union MO 1988 r < ab/( 2(a + b) ) inradius of tetrahedron

ASU 477 All Soviet Union MO 1988 min of b/(c + d)+c/(a + b) if b+c>= a + d

ASU 472 All Soviet Union MO 1988 2(sin A)/A +... <= (1/B + 1/C) sin A +...

ASU 469 All Soviet Union MO 1988 x^5+y^5=2x^2y^2 =>(1-xy) rational's square

ASU 468 All Soviet Union MO 1988 blackboard game (m and n)=> m+n+mn

ASU 466 All Soviet Union MO 1988 19 pos. <=88, 88 pos. <=19, same sum