MathDB

1

MMO 274 Moscow MO 1954 7x7 linear system

\begin{cases} 10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\ 11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\ 15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\ 2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\ 6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\ 3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\ 4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}

281

1

MMO 281 Moscow MO 1954 sum x_i^2 >10000, sum x_1<300

*. Positive numbers

x_1, x_2, ..., x_{100}

satisfy the system

\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ x_1 + x_2 + ...+ x_{100} < 300 \end{cases}

Prove that among these numbers there are three whose sum is greater than

100

.

282

1

MMO 282 Moscow MO 1954 sequence 1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13

Given a sequence of numbers

a_1, a_2, ..., a_{15}

, one can always construct a new sequence

b_1,b_2, ..., b_{15}

, where

b_i

is equal to the number of terms in the sequence

\{a_k\}^{15}_{k=1}

less than

a_i

(

i = 1, 2,..., 15

). Is there a sequence

\{a_k\}^{15}_{k=1}

for which the sequence

\{b_k\}^{15}_{k=1}

is

1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13 \,?

285

1

MMO 285 Moscow MO 1954 x^2+Ax+B=0, x^2+Cx+D=0, x^2+(A+C)/2 x+(B+D)/2=0

The absolute values of all roots of the quadratic equation

x^2+Ax+B = 0

and

x^2+Cx+D = 0

are less then

1

. Prove that so are absolute values of the roots of the quadratic equation

x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0

.

286

1

MMO 286 Moscow MO 1954 10-digit numbers, figures 1 and 2, subsets

Consider the set of all

10

-digit numbers expressible with the help of figures

1

and

2

only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two

3

’s.

283

1

MMO 283 Moscow MO 1954 1001 free endpoints, 5 segments

Consider five segments

AB_1, AB_2, AB_3, AB_4, AB_5

. From each point

B_i

there can exit either

5

segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to

1001

? (A free endpoint is an endpoint from which no segment begins.)

284

1

MMO 284 Moscow MO 1954 planes of symmetry a triangular pyramid may have

How many planes of symmetry can a triangular pyramid have?

280

1

MMO 280 Moscow MO 1954 locus of line's intersections

Rays

l_1

and

l_2

pass through a point

O

. Segments

OA_1

and

OB_1

on

l_1

, and

OA_2

and

OB_2

on

l_2

, are drawn so that

\frac{OA_1}{OA_2} \ne \frac{OB_1}{OB_2}

. Find the set of all intersection points of lines

A_1A_2

and

B_1B_2

as

l_2

rotates around

O

while

l_1

is fixed.

279

1

MMO 279 Moscow MO 1954 OB <= OA_1/ 2, starting with 4 concurrent lines

Given four straight lines,

m_1, m_2, m_3, m_4

, intersecting at

O

and numbered clockwise with

O

as the center of the clock, we draw a line through an arbitrary point

A_1

on

m_1

parallel to

m_4

until the line meets

m_2

at

A_2

. We draw a line through

A_2

parallel to

m_1

until it meets

m_3

at

A_3

. We also draw a line through

A_3

parallel to

m_2

until it meets

m_4

at

A_4

. Now, we draw a line through

A_4

parallel to

m_3

until it meets

m_1

at

B

. Prove that
a)

OB< \frac{OA_1}{2}

.
b)

OB \le \frac{OA_1}{4}

. https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png

278

1

MMO 278 Moscow MO 1954 no from 1 to 70 in 17x17 square table, AB // CD

A

17 \times 17

square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from

1

to

70

. Prove that there are

4

distinct squares whose centers

A, B, C, D

are the vertices of a parallelogramsuch that

AB // CD

, moreover, the sum of the numbers in the squares with centers

A

and

C

is equal to that in the squares with centers

B

and

D

.

277

1

MMO 277 Moscow MO 1954 map of town with equilateral triangles

The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections;

6

streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points

A

and

B

situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through

120^o

to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.) https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png

276

1

MMO 276 Moscow MO 1954 divisors of product of primes 2 to 31, combo sum

a) Let

1, 2, 3, 5, 6, 7, 10, .., N

be all the divisors of

N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31

(the product of primes

2

to

31

) written in increasing order. Below this series of divisors, write the following series of

1

’s or

-1

’s: write

1

below any number that factors into an even number of prime factors and below a

1

, write

-1

below the remaining numbers. Prove that the sum of the series of

1

’s and

-1

’s is equal to

0

.
b) Let

1, 2, 3, 5, 6, 7, 10, .., N

be all the divisors of

N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37

(the product of primes

2

to

37

) written in increasing order. Below this series of divisors, write the following series of

1

’s or

-1

’s: write

1

below any number that factors into an even number of prime factors and below a

1

, write

-1

below the remaining numbers. Prove that the sum of the series of

1

’s and

-1

’s is equal to

0

.

275

1

MMO 275 Moscow MO 1954 no axes of symmetry a heptagon can have

How many axes of symmetry can a heptagon have?

273

1

MMO 273 Moscow MO 1954 letters on a piece of graph paper, picture

Given a piece of graph paper with a letter assigned to each vertex of every square such that on every segment connecting two vertices that have the same letter and are on the same line of the mesh, there is at least one vertex with another letter. What is the least number of distinct letters needed to plot such a picture, along the sides of the cells?

267

1

MMO 267 Moscow MO 1954 (x - x_0)^2/ (x^4+a_1x^3+a_2x^2+a_3x+a_4)

Prove that if

x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,

then

x^4 + a_1x^3 + a_2x^2 + a_3x + a_4

is a mutliple of

(x - x_0)^2

.

269

1

MMO 269 Moscow MO 1954 system inequalities 100x100

a) Given

100

numbers

a_1, ..., a_{100}

such that

\begin{cases} a_1 - 3a_2 + 2a_3 \ge 0, \\ a_2 - 3a_3 + 2a_4 \ge 0, \\ a_3 - 3a_4 + 2a_5 \ge 0, \\ ... \\ a_{99} - 3a_{100} + 2a_1 \ge 0, \\ a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}

prove that the numbers are equal.
b) Given numbers

a_1=1, ..., a_{100}

such that

\begin{cases} a_1 - 4a_2 + 3a_3 \ge 0, \\ a_2 - 4a_3 + 3a_4 \ge 0, \\ a_3 - 4a_4 + 3a_5 \ge 0, \\ ... \\ a_{99} - 4a_{100} + 3a_1 \ge 0, \\ a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}

Find

a_2, a_3, ... , a_{100}.

272

1

MMO 272 Moscow MO 1954 x^2 + 2x sin (xy) + 1 = 0

Find all real solutions of the equation

x^2 + 2x \sin (xy) + 1 = 0

.

271

1

MMO 271 Moscow MO 1954 AB=CD=8, AC=BD=10, AD=BC=13 in space

Do there exist points

A, B, C, D

in space, such that

AB = CD = 8, AC = BD = 10

, and

AD = BC = 13

?

270

1

MMO 270 Moscow MO 1954 one of 4 quadrilaterals has both angles not acute

Consider

\vartriangle ABC

and a point

S

inside it. Let

A_1, B_1, C_1

be the intersection points of

AS, BS, CS

with

BC, AC, AB

, respectively. Prove that at least in one of the resulting quadrilaterals

AB_1SC_1, C_1SA_1B, A_1SB_1C

both angles at either

C_1

and

B_1

, or

C_1

and

A_1

, or

A_1

and

B_1

are not acute.

268

1

MMO 268 Moscow MO 1954 1234567891011... 9899100 delete 100 digits

Delete

100

digits from the number

1234567891011... 9899100

so that the remaining number were as big as possible.

266

1

MMO 266 Moscow MO 1954 system x(1-1/2^n)+y(1-1/(2^{n+1}))+...

Find all solutions of the system consisting of

3

equations:

x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0

for

n = 1, 2, 3

.

265

1

MMO 265 Moscow MO 1954 sum of vectors in a convex n-gon

From an arbitrary point

O

inside a convex

n

-gon, perpendiculars are drawn on (extensions of the) sides of the

n

-gon. Along each perpendicular a vector is constructed, starting from

O

, directed towards the side onto which the perpendicular is drawn, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

264

1

MMO 264 Moscow MO 1954 cut a 3&times;3 square unfolding cube edge 1

* Cut out of a

3 \times 3

square an unfolding of the cube with edge

1

.

263

1

MMO 263 Moscow MO 1954 max ratio 3-digit to sum of it's digits

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

262

1

MMO 262 Moscow MO 1954 diophantine m^2 + 1954 = n^2

Are there integers

m

and

n

such that

m^2 + 1954 = n^2

?

261

1

MMO 261 Moscow MO 1954 number theory, divisiond by 1 digit numbers

Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines: https://cdn.artofproblemsolving.com/attachments/2/a/e1d3c68ec52e11ad59de755c3dbdc2cf54a81f.png

260

1

MMO 260 Moscow MO 1954 convex polygons with equal sides and n-3 angles

Given two convex polygons,

A_1A_2...A_n

and

B_1B_2...B_n

such that

A_1A_2 = B_1B_2

,

A_2A_3 = B_2B_3

,

...

,

A_nA_1 = B_nB_1

and

n - 3

angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.

259

1

MMO 259 Moscow MO 1954 regular star-shaped hexagon split in 4 parts

A regular star-shaped hexagon is split into

4

parts. Construct from them a convex polygon.
Note: A regular six-pointed star is a figure that is obtained by combining a regular triangle and a triangle symmetrical to it relative to its center

1954 Moscow Mathematical Olympiad

Subcontests

MMO 274 Moscow MO 1954 7x7 linear system

MMO 281 Moscow MO 1954 sum x_i^2 >10000, sum x_1<300

MMO 282 Moscow MO 1954 sequence 1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13

MMO 285 Moscow MO 1954 x^2+Ax+B=0, x^2+Cx+D=0, x^2+(A+C)/2 x+(B+D)/2=0

MMO 286 Moscow MO 1954 10-digit numbers, figures 1 and 2, subsets

MMO 283 Moscow MO 1954 1001 free endpoints, 5 segments

MMO 284 Moscow MO 1954 planes of symmetry a triangular pyramid may have

MMO 280 Moscow MO 1954 locus of line's intersections

MMO 279 Moscow MO 1954 OB <= OA_1/ 2, starting with 4 concurrent lines

MMO 278 Moscow MO 1954 no from 1 to 70 in 17x17 square table, AB // CD

MMO 277 Moscow MO 1954 map of town with equilateral triangles

MMO 276 Moscow MO 1954 divisors of product of primes 2 to 31, combo sum

MMO 275 Moscow MO 1954 no axes of symmetry a heptagon can have

MMO 273 Moscow MO 1954 letters on a piece of graph paper, picture

MMO 267 Moscow MO 1954 (x - x_0)^2/ (x^4+a_1x^3+a_2x^2+a_3x+a_4)

MMO 269 Moscow MO 1954 system inequalities 100x100

MMO 272 Moscow MO 1954 x^2 + 2x sin (xy) + 1 = 0

MMO 271 Moscow MO 1954 AB=CD=8, AC=BD=10, AD=BC=13 in space

MMO 270 Moscow MO 1954 one of 4 quadrilaterals has both angles not acute

MMO 268 Moscow MO 1954 1234567891011... 9899100 delete 100 digits

MMO 266 Moscow MO 1954 system x(1-1/2^n)+y(1-1/(2^{n+1}))+...

MMO 265 Moscow MO 1954 sum of vectors in a convex n-gon

MMO 264 Moscow MO 1954 cut a 3&times;3 square unfolding cube edge 1

MMO 263 Moscow MO 1954 max ratio 3-digit to sum of it's digits

MMO 262 Moscow MO 1954 diophantine m^2 + 1954 = n^2

MMO 261 Moscow MO 1954 number theory, divisiond by 1 digit numbers

MMO 260 Moscow MO 1954 convex polygons with equal sides and n-3 angles

MMO 259 Moscow MO 1954 regular star-shaped hexagon split in 4 parts

1954 Moscow Mathematical Olympiad

Subcontests

MMO 274 Moscow MO 1954 7x7 linear system

MMO 281 Moscow MO 1954 sum x_i^2 &gt;10000, sum x_1&lt;300

MMO 282 Moscow MO 1954 sequence 1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13

MMO 285 Moscow MO 1954 x^2+Ax+B=0, x^2+Cx+D=0, x^2+(A+C)/2 x+(B+D)/2=0

MMO 286 Moscow MO 1954 10-digit numbers, figures 1 and 2, subsets

MMO 283 Moscow MO 1954 1001 free endpoints, 5 segments

MMO 284 Moscow MO 1954 planes of symmetry a triangular pyramid may have

MMO 280 Moscow MO 1954 locus of line's intersections

MMO 279 Moscow MO 1954 OB &lt;= OA_1/ 2, starting with 4 concurrent lines

MMO 278 Moscow MO 1954 no from 1 to 70 in 17x17 square table, AB // CD

MMO 277 Moscow MO 1954 map of town with equilateral triangles

MMO 276 Moscow MO 1954 divisors of product of primes 2 to 31, combo sum

MMO 275 Moscow MO 1954 no axes of symmetry a heptagon can have

MMO 273 Moscow MO 1954 letters on a piece of graph paper, picture

MMO 267 Moscow MO 1954 (x - x_0)^2/ (x^4+a_1x^3+a_2x^2+a_3x+a_4)

MMO 269 Moscow MO 1954 system inequalities 100x100

MMO 272 Moscow MO 1954 x^2 + 2x sin (xy) + 1 = 0

MMO 271 Moscow MO 1954 AB=CD=8, AC=BD=10, AD=BC=13 in space

MMO 270 Moscow MO 1954 one of 4 quadrilaterals has both angles not acute

MMO 268 Moscow MO 1954 1234567891011... 9899100 delete 100 digits

MMO 266 Moscow MO 1954 system x(1-1/2^n)+y(1-1/(2^{n+1}))+...

MMO 265 Moscow MO 1954 sum of vectors in a convex n-gon

MMO 264 Moscow MO 1954 cut a 3&amp;times;3 square unfolding cube edge 1

MMO 263 Moscow MO 1954 max ratio 3-digit to sum of it's digits

MMO 262 Moscow MO 1954 diophantine m^2 + 1954 = n^2

MMO 261 Moscow MO 1954 number theory, divisiond by 1 digit numbers

MMO 260 Moscow MO 1954 convex polygons with equal sides and n-3 angles

MMO 259 Moscow MO 1954 regular star-shaped hexagon split in 4 parts

MMO 281 Moscow MO 1954 sum x_i^2 >10000, sum x_1<300

MMO 279 Moscow MO 1954 OB <= OA_1/ 2, starting with 4 concurrent lines

MMO 264 Moscow MO 1954 cut a 3×3 square unfolding cube edge 1