MathDB

1

MMO 372 Moscow MO 1957 divide a_i <= a_{i+1} <= 2a_i, into 2 groups

n

integers

a_1 = 1, a_2,..., a_n

such that

a_i \le a_{i+1} \le 2a_i

(

i = 1, 2, 3,..., n - 1

) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

371

1

MMO 371 Moscow MO 1957 inscribe a rectangle in given ABCD

Given quadrilateral

ABCD

and the directions of its sides. Inscribe a rectangle in

ABCD

.

370

1

MMO 370 Moscow MO 1957 3 equal circles, tangent in pairns

* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.

369

1

MMO 369 Moscow MO 1957 1957 as sum of 12 pos. integer , min prod. a_i!

Represent

1957

as the sum of

12

positive integer summands

a_1, a_2, ... , a_{12}

for which the number

a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!

is minimal.

368

1

MMO 368 Moscow MO 1957 system 99x99 ... 1-x_1^2=x_2

Find all real solutions of the system : (a)

\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}

(b)*

\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}

367

1

MMO 367 Moscow MO 1957 intersection of 2 squares in 8 points

Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.

366

1

MMO 366 Moscow MO 1957 system 3x3 .. 2x_1^2/(1+x_1^2)=x_2

Solve the system:

\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}

365

1

MMO 365 Moscow MO 1957 A'O/A'G + B'O/B'G + C'O/C'G = 3

(a) Given a point

O

inside an equilateral triangle

\vartriangle ABC

. Line

OG

connects

O

with the center of mass

G

of the triangle and intersects the sides of the triangle, or their extensions, at points

A', B', C'

. Prove that

\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.

(b) Point

G

is the center of the sphere inscribed in a regular tetrahedron

ABCD

. Straight line

OG

connecting

G

with a point

O

inside the tetrahedron intersects the faces at points

A', B', C', D'

. Prove that

\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.

364

1

MMO 364 Moscow MO 1957 no of digits 1,2,3,...,10&8 equals 0s in 1,2,3, ...,10^9

(a) Prove that the number of all digits in the sequence

1, 2, 3,... , 10^8

is equal to the number of all zeros in the sequence

1, 2, 3, ... , 10^9

.
(b) Prove that the number of all digits in the sequence

1, 2, 3, ... , 10^k

is equal to the number of all zeros in the sequence

1, 2, 3, ... , 10^{k+1}

.

363

1

MMO 363 Moscow MO 1957 sum of 8 consecutive Fibonacci

Eight consecutive numbers are chosen from the Fibonacci sequence

1, 2, 3, 5, 8, 13, 21,...

. Prove that the sequence does not contain the sum of chosen numbers.

362

1

MMO 362 Moscow MO 1957 sequence of triangles in/circum-scribed in circle

(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles.
(b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

361

1

MMO 361 Moscow MO 1957 min largest angle of triangle with given sides a,b

The lengths,

a

and

b

, of two sides of a triangle are known.
(a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value?
(b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?

360

1

MMO 360 Moscow MO 1957 radio lamp has a 20-contact plug, contacts in circle

(a) A radio lamp has a

7

-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with

7

holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number?
(b) A radio lamp has a

20

-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with

20

holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?

359

1

MMO 359 Moscow MO 1957 locus of endpoints M of all broken lines OM

Straight lines

OA

and

OB

are perpendicular. Find the locus of endpoints

M

of all broken lines

OM

of length

\ell

which intersect each line parallel to

OA

or

OB

at not more than one point.

358

1

MMO 358 Moscow MO 1957 closed broken line in space, no of segments div. by 6

The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by

6

.

357

1

MMO 357 Moscow MO 1957 (20^n + 16^n - 3^n - 1) divisible by 323

For which integer

n

is

N = 20^n + 16^n - 3^n - 1

divisible by

323

?

356

1

MMO 356 Moscow MO 1957 polygon made of rigid rods connected by hinges

A planar polygon

A_1A_2A_3 . . .A_{n-1}A_n

(

n > 4

) is made of rigid rods that are connected by hinges. Is it possible to bend the polygon (at hinges only!) into a triangle?

355

1

MMO 355 Moscow MO 1957 student uses subway and tram, money

a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change. b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change.
Note: In

1957

, the price of a subway ticket was

50

kopeks, that of a tram ticket

30

kopeks, the denominations of the coins were

1, 2, 3, 5, 10, 15

, and

20

kopeks. (

1

rouble =

100

kopeks.)

354

1

MMO 354 Moscow MO 1957 equal segments lead to parallel sides

In a quadrilateral

ABCD

points

M

and

N

are the midpoints of the diagonals

AC

and

BD

, respectively. The line through

M

and

N

meets

AB

and

CD

at

M'

and

N'

, respectively. Prove that if

MM' = NN'

, then

AD // BC

.

353

1

MMO 353 Moscow MO 1957 floor function equation x^3 - [x] = 3 ,

Solve the equation

x^3 - [x] = 3

.

352

1

MMO 352 Moscow MO 1957 parallelogram. min diagonal , fixed area

Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

349

1

MMO 349 Moscow MO 1957 sum 1 or all numbers=0 in a rectangle table

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to

1

or all the numbers are equal to

0

.

351

1

MMO 351 Moscow MO 1957 locus of 4th vertex of rectangle

Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.

350

1

MMO 350 Moscow MO 1957 signs of distance between 2 towns, using 2 digits

The distance between towns

A

and

B

is

999

km. At every kilometer of the road that connects

A

and

B

a sign shows the distances to

A

and

B

as follows:

\fbox{0-999}

,

\fbox{1-998}

,

\fbox{2-997}

,

. . .

,

\fbox{998-1}

,

\fbox{999-0}

How many signs are there, with both distances written with the help of only two distinct digits?

348

1

MMO 348 Moscow MO 1957 snail crawls over table at constant speed

A snail crawls over a table at a constant speed. Every

15

minutes it turns by

90^o

, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

347

1

MMO 347 Moscow MO 1957 5/ (ax^3+bx^2+cx+d) for any integer x => 5 / a,b,c,d

a) Let

ax^3 + bx^2 + cx + d

be divisible by

5

for given positive integers

a, b, c, d

and any integer

x

. Prove that

a, b, c

and

d

are all divisible by

5

.
b) Let

ax^4 + bx^3 + cx^2 + dx + e

be divisible by

7

for given positive integers

a, b, c, d, e

and all integers

x

. Prove that

a, b, c, d

and

e

are all divisible by

7

.

346

1

MMO 346 Moscow MO 1957 isoceles trapezoid divided into isosceles, diagonal

Find all isosceles trapezoids that are divided into

2

isosceles triangles by a diagonal.

1957 Moscow Mathematical Olympiad

Subcontests

MMO 372 Moscow MO 1957 divide a_i <= a_{i+1} <= 2a_i, into 2 groups

MMO 371 Moscow MO 1957 inscribe a rectangle in given ABCD

MMO 370 Moscow MO 1957 3 equal circles, tangent in pairns

MMO 369 Moscow MO 1957 1957 as sum of 12 pos. integer , min prod. a_i!

MMO 368 Moscow MO 1957 system 99x99 ... 1-x_1^2=x_2

MMO 367 Moscow MO 1957 intersection of 2 squares in 8 points

MMO 366 Moscow MO 1957 system 3x3 .. 2x_1^2/(1+x_1^2)=x_2

MMO 365 Moscow MO 1957 A'O/A'G + B'O/B'G + C'O/C'G = 3

MMO 364 Moscow MO 1957 no of digits 1,2,3,...,10&8 equals 0s in 1,2,3, ...,10^9

MMO 363 Moscow MO 1957 sum of 8 consecutive Fibonacci

MMO 362 Moscow MO 1957 sequence of triangles in/circum-scribed in circle

MMO 361 Moscow MO 1957 min largest angle of triangle with given sides a,b

MMO 360 Moscow MO 1957 radio lamp has a 20-contact plug, contacts in circle

MMO 359 Moscow MO 1957 locus of endpoints M of all broken lines OM

MMO 358 Moscow MO 1957 closed broken line in space, no of segments div. by 6

MMO 357 Moscow MO 1957 (20^n + 16^n - 3^n - 1) divisible by 323

MMO 356 Moscow MO 1957 polygon made of rigid rods connected by hinges

MMO 355 Moscow MO 1957 student uses subway and tram, money

MMO 354 Moscow MO 1957 equal segments lead to parallel sides

MMO 353 Moscow MO 1957 floor function equation x^3 - [x] = 3 ,

MMO 352 Moscow MO 1957 parallelogram. min diagonal , fixed area

MMO 349 Moscow MO 1957 sum 1 or all numbers=0 in a rectangle table

MMO 351 Moscow MO 1957 locus of 4th vertex of rectangle

MMO 350 Moscow MO 1957 signs of distance between 2 towns, using 2 digits

MMO 348 Moscow MO 1957 snail crawls over table at constant speed

MMO 347 Moscow MO 1957 5/ (ax^3+bx^2+cx+d) for any integer x => 5 / a,b,c,d

MMO 346 Moscow MO 1957 isoceles trapezoid divided into isosceles, diagonal

1957 Moscow Mathematical Olympiad

Subcontests

MMO 372 Moscow MO 1957 divide a_i &lt;= a_{i+1} &lt;= 2a_i, into 2 groups

MMO 371 Moscow MO 1957 inscribe a rectangle in given ABCD

MMO 370 Moscow MO 1957 3 equal circles, tangent in pairns

MMO 369 Moscow MO 1957 1957 as sum of 12 pos. integer , min prod. a_i!

MMO 368 Moscow MO 1957 system 99x99 ... 1-x_1^2=x_2

MMO 367 Moscow MO 1957 intersection of 2 squares in 8 points

MMO 366 Moscow MO 1957 system 3x3 .. 2x_1^2/(1+x_1^2)=x_2

MMO 365 Moscow MO 1957 A&#039;O/A&#039;G + B&#039;O/B&#039;G + C&#039;O/C&#039;G = 3

MMO 364 Moscow MO 1957 no of digits 1,2,3,...,10&amp;8 equals 0s in 1,2,3, ...,10^9

MMO 363 Moscow MO 1957 sum of 8 consecutive Fibonacci

MMO 362 Moscow MO 1957 sequence of triangles in/circum-scribed in circle

MMO 361 Moscow MO 1957 min largest angle of triangle with given sides a,b

MMO 360 Moscow MO 1957 radio lamp has a 20-contact plug, contacts in circle

MMO 359 Moscow MO 1957 locus of endpoints M of all broken lines OM

MMO 358 Moscow MO 1957 closed broken line in space, no of segments div. by 6

MMO 357 Moscow MO 1957 (20^n + 16^n - 3^n - 1) divisible by 323

MMO 356 Moscow MO 1957 polygon made of rigid rods connected by hinges

MMO 355 Moscow MO 1957 student uses subway and tram, money

MMO 354 Moscow MO 1957 equal segments lead to parallel sides

MMO 353 Moscow MO 1957 floor function equation x^3 - [x] = 3 ,

MMO 352 Moscow MO 1957 parallelogram. min diagonal , fixed area

MMO 349 Moscow MO 1957 sum 1 or all numbers=0 in a rectangle table

MMO 351 Moscow MO 1957 locus of 4th vertex of rectangle

MMO 350 Moscow MO 1957 signs of distance between 2 towns, using 2 digits

MMO 348 Moscow MO 1957 snail crawls over table at constant speed

MMO 347 Moscow MO 1957 5/ (ax^3+bx^2+cx+d) for any integer x =&gt; 5 / a,b,c,d

MMO 346 Moscow MO 1957 isoceles trapezoid divided into isosceles, diagonal

MMO 372 Moscow MO 1957 divide a_i <= a_{i+1} <= 2a_i, into 2 groups

MMO 365 Moscow MO 1957 A'O/A'G + B'O/B'G + C'O/C'G = 3

MMO 364 Moscow MO 1957 no of digits 1,2,3,...,10&8 equals 0s in 1,2,3, ...,10^9

MMO 347 Moscow MO 1957 5/ (ax^3+bx^2+cx+d) for any integer x => 5 / a,b,c,d