MathDB

1

MMO 313- Moscow MO 1955 2 intersecting lines and a transfromation

Given two lines in a plane, intersecting at an acute angle. In the direction of one of the straight lines, compression is performed with a coefficient of 1/2. Prove that there is a point from which the distance to the point of intersection of the lines increases.
Note: What is meant here is a transformation in which each point moves parallel to one straight line so that its distance to the second straight line is halved, while it remains the same side from the second straight line.
[hide=original wording] На плоскости даны две прямые, пересекающиеся под острым углом. В направлении одной из прямых производится сжатие 1 с коэффициентом 1/2. Доказать, что найдется точка, расстояние от которой до точки пересечения прямых увеличится.
Здесь имеется в виду преобразование, при котором каждая точка перемещается параллельно одной прямой так, что её расстояние до второй прямой уменьшается вдвое, причём она остаётся по ту же самую сторону от второй прямой

319

1

MMO 319 Moscow MO 1955 triangles inscribed in triangles, equal ratio

Consider

\vartriangle A_0B_0C_0

and points

C_1, A_1, B_1

on its sides

A_0B_0, B_0C_0, C_0A_0

, points

C_2, A_2,B_2

on the sides

A_1B_1, B_1C_1, C_1A_1

of

\vartriangle A_1B_1C_1

, respectively, etc., so that

\frac{A_0B_1}{B_1C_0}= \frac{B_0C_1}{C_1A_0}= \frac{C_0A_1}{A_1B_0}= k, \frac{A_1B_2}{B_2C_1}= \frac{B_1C_2}{C_2A_1}= \frac{C_1A_2}{A_2B_1}= \frac{1}{k^2}

and, in general,

\frac{A_nB_{n+1}}{B_{n+1}C_n}= \frac{B_nC_{n+1}}{C_{n+1}A_n}= \frac{C_nA_{n+1}}{A_{n+1}B_n} =k^{2n}

for

n

even ,

\frac{1}{k^{2n}}

for

n

odd. Prove that

\vartriangle ABC

formed by lines

A_0A_1, B_0B_1, C_0C_1

is contained in

\vartriangle A_nB_nC_n

for any

n

.

318

1

MMO 318 Moscow MO 1955 max no of triples from 1955 points, each 2 1 common

What greatest number of triples of points can be selected from

1955

given points, so that each two triples have one common point?

317

1

MMO 317 Moscow MO 1955 source of light on right circular cone

A right circular cone stands on plane

P

. The radius of the cone’s base is

r

, its height is

h

. A source of light is placed at distance

H

from the plane, and distance

1

from the axis of the cone. What is the illuminated part of the disc of radius

R

, that belongs to

P

and is concentric with the disc forming the base of the cone?

316

1

MMO 316 Moscow MO 1955 p/q root of integer f(x) => (p - kq) /f(k)

Prove that if

\frac{p}{q}

is an irreducible rational number that serves as a root of the polynomial

f(x) = a_0x^n + a_1x^{n-1} + ... + a_n

with integer coefficients, then

p - kq

is a divisor of

f(k)

for any integer

k

.

315

1

MMO 315 Moscow MO 1955 5 men play dominoes, partner, opponent

Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

314

1

MMO 314 Moscow MO 1955 x^n - a_1x^{n-1} -...-a_o=0, a_i>=0

Prove that the equation

x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0

, where

a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0

, cannot have two positive roots.

313

1

MMO 313 Moscow MO 1955 system of segments on line, arithm. progression

On the numerical line, arrange a system of closed segments of length

1

without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.

312

1

MMO 312 Moscow MO 1955 equal rations lead to parallels, Δ inscribed in Δ

Given

\vartriangle ABC

, points

C_1, A_1, B_1

on sides

AB, BC, CA

, respectively, such that

\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n}

and points

C_2, A_2, B_2

on sides

A_1B_1, B_1C_1, C_1A_1

of

\vartriangle A_1B_1C_1

, respectively, such that

\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n

. Prove that

A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA

.

311

1

MMO 311 Moscow MO 1955 [a], .., [Na] and [1/a],...[N/a], are distinct

Find all numbers

a

such that
(1) all numbers

[a], [2a], . . . , [Na]

are distinct and
(2) all numbers

\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]

are distinct.

310

1

MMO 310 Moscow MO 1955 given 6-variable inequality, triangle sides exist?

Let the inequality

Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}

with given

a > 0, b > 0, c > 0

hold for all

A > 0, B > 0, C > 0

. Is it possible to construct a triangle with sides of lengths

a, b, c

?

309

1

MMO 309 Moscow MO 1955 numbering sides of n-gon

A point

O

inside a convex

n

-gon

A_1A_2 . . .A_n

is connected with segments to its vertices. The sides of this

n

-gon are numbered

1

to

n

(distinct sides have distinct numbers). The segments

OA_1,OA_2, . . . ,OA_n

are similarly numbered.
a) For

n = 9

find a numeration such that the sum of the sides’ numbers is the same for all triangles

A_1OA_2, A_2OA_3, . . . , A_nOA_1

.
b) Prove that for

n = 10

there is no such numeration.

308

1

MMO 308 Moscow MO 1955 3 tangent circlee, inner tangent bisects arc

* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.

307

1

MMO 307 Moscow MO 1955 ax^2+bx+c= square for any integer x

* The quadratic expression

ax^2 + bx + c

is a square (of an integer) for any integer

x

. Prove that

ax^2 + bx + c = (dx + e)^2

for some integers d and e.

306

1

MMO 306 Moscow MO 1955 cut a rectangle into 18 rectangles

Cut a rectangle into

18

rectangles so that no two adjacent ones form a rectangle.

305

1

MMO 305 Moscow MO 1955 25 in a chess tournament

25

chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?

304

1

MMO 304 Moscow MO 1955 triangle of excenters is acute

The centers

O_1, O_2

and

O_3

of circles exscribed about

\vartriangle ABC

are connected. Prove that

O_1O_2O_3

is an acute-angled one.

303

1

MMO 303 Moscow MO 1955 ax^2+bx+c = 4-th power of integer x

The quadratic expression

ax^2+bx+c

is the

4

-th power (of an integer) for any integer

x

. Prove that

a = b = 0

.

302

1

MMO 302 Moscow MO 1955 diophantine x^3 - 2y^3 - 4z^3 = 0

Find integer solutions of the equation

x^3 - 2y^3 - 4z^3 = 0

.

301

1

MMO 301 Moscow MO 1955 <OAB, <OBA, <OBC, <OCB,<OAC,<OCA acute, trihedral

Given a trihedral angle with vertex

O

. Find whether there is a planar section

ABC

such that the angles

\angle OAB

,

\angle OBA

,

\angle OBC

,

\angle OCB

,

\angle OAC

,

\angle OCA

are acute.

300

1

MMO 300 Moscow MO 1955 AC - DA > 1, BC - BD > 1, EC - ED > 1

Inside

\vartriangle ABC

, there is fixed a point

D

such that

AC - DA > 1

and

BC - BD > 1

. Prove that

EC - ED > 1

for any point

E

on segment

AB

.

299

1

MMO 299 Moscow MO 1955 increasing arithmetic progression of primes

Suppose that primes

a_1, a_2, . . . , a_p

form an increasing arithmetic progression and

a_1 > p

. Prove that if

p

is a prime, then the difference of the progression is divisible by

p

.

298

1

MMO 298 Moscow MO 1955 system x^3 + y^3 = 1, x^4 + y^4 = 1

Find all real solutions of the system

\begin{cases} x^3 + y^3 = 1 \\ x^4 + y^4 = 1 \end{cases}

297

1

MMO 297 Moscow MO 1955 locus of midpoints of segments on 2 circles

Given two distinct nonintersecting circles none of which is inside the other. Find the locus of the midpoints of all segments whose endpoints lie on the circles.

296

1

MMO 296 Moscow MO 1955 cyclic ABCD, 4 circles, concyclic intersections

There are four points

A, B, C, D

on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by

A_1, B_1, C_1, D_1

. (Some of these points may coincide with previously given ones.) Prove that points

A_1, B_1, C_1, D_1

lie on one circle.

295

1

MMO 295 Moscow MO 1955 convex domains on plane contains straight line

Which convex domains (figures) on a plane can contain an entire straight line?
It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

294

1

MMO 294 Moscow MO 1955 square table witn no 1-49, symmetric

a) A square table with

49

small squares is filled with numbers

1

to

7

so that in each row and in each column all numbers from

1

to

7

are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers

1, 2, 3, . . . , 7

are present.
b) A square table with

n^2

small squares is filled with numbers

1

to

n

so that in each row and in each column all numbers from

1

to

n

are present. Let

n

be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers

1, 2, 3, . . . , n

are present.

293

1

MMO 293 Moscow MO 1955 locus, equal segments on sides of ABCD

Consider a quadrilateral

ABCD

and points

K, L, M, N

on sides

AB, BC, CD

and

AD

, respectively, such that

KB = BL = a, MD = DN = b

and

KL \nparallel MN

. Find the set of all the intersection points of

KL

with

MN

as

a

and

b

vary.

292

1

MMO 292 Moscow MO 1955 2^n = 10a + b , 6 \ ab

Let

a, b, n

be positive integers,

b < 10

and

2^n = 10a + b

. Prove that if

n > 3

, then

6

divides

ab

.

291

1

MMO 291 Moscow MO 1955 rectangles cut into 13 equal squares

Find all rectangles that can be cut into

13

equal squares.

290

1

MMO 290 Moscow MO 1955 n^2 + n + 1 divisible by 1955

Is there an integer

n

such that

n^2 + n + 1

is divisible by

1955

?

289

1

MMO 289 Moscow MO 1955 locus, related to equilateral and AE=CD

Consider an equilateral triangle

\vartriangle ABC

and points

D

and

E

on the sides

AB

and

BC

csuch that

AE = CD

. Find the locus of intersection points of

AE

with

CD

as points

D

and

E

vary.

288

1

MMO 288 Moscow MO 1955 area wanted, circle bisects segment, right triangle

We are given a right triangle

ABC

and the median

BD

drawn from the vertex

B

of the right angle. Let the circle inscribed in

\vartriangle ABD

be tangent to side

AD

at

K

. Find the angles of

\vartriangle ABC

if

K

divides

AD

in halves.

287

1

MMO 287 Moscow MO 1955 square table with numbers 1 to 49, sum

a) The numbers

1, 2, . . . , 49

are arranged in a square table as follows:
https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png
Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of

36

numbers, etc.,

7

times. Find the sum of the numbers selected.
b) The numbers

1, 2, . . . , k^2

are arranged in a square table as follows:
https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png
Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of

(k - 1)^2

numbers, etc.,

k

times. Find the sum of the numbers selected.

1955 Moscow Mathematical Olympiad

Subcontests

MMO 313- Moscow MO 1955 2 intersecting lines and a transfromation

MMO 319 Moscow MO 1955 triangles inscribed in triangles, equal ratio

MMO 318 Moscow MO 1955 max no of triples from 1955 points, each 2 1 common

MMO 317 Moscow MO 1955 source of light on right circular cone

MMO 316 Moscow MO 1955 p/q root of integer f(x) => (p - kq) /f(k)

MMO 315 Moscow MO 1955 5 men play dominoes, partner, opponent

MMO 314 Moscow MO 1955 x^n - a_1x^{n-1} -...-a_o=0, a_i>=0

MMO 313 Moscow MO 1955 system of segments on line, arithm. progression

MMO 312 Moscow MO 1955 equal rations lead to parallels, Δ inscribed in Δ

MMO 311 Moscow MO 1955 [a], .., [Na] and [1/a],...[N/a], are distinct

MMO 310 Moscow MO 1955 given 6-variable inequality, triangle sides exist?

MMO 309 Moscow MO 1955 numbering sides of n-gon

MMO 308 Moscow MO 1955 3 tangent circlee, inner tangent bisects arc

MMO 307 Moscow MO 1955 ax^2+bx+c= square for any integer x

MMO 306 Moscow MO 1955 cut a rectangle into 18 rectangles

MMO 305 Moscow MO 1955 25 in a chess tournament

MMO 304 Moscow MO 1955 triangle of excenters is acute

MMO 303 Moscow MO 1955 ax^2+bx+c = 4-th power of integer x

MMO 302 Moscow MO 1955 diophantine x^3 - 2y^3 - 4z^3 = 0

MMO 301 Moscow MO 1955 <OAB, <OBA, <OBC, <OCB,<OAC,<OCA acute, trihedral

MMO 300 Moscow MO 1955 AC - DA > 1, BC - BD > 1, EC - ED > 1

MMO 299 Moscow MO 1955 increasing arithmetic progression of primes

MMO 298 Moscow MO 1955 system x^3 + y^3 = 1, x^4 + y^4 = 1

MMO 297 Moscow MO 1955 locus of midpoints of segments on 2 circles

MMO 296 Moscow MO 1955 cyclic ABCD, 4 circles, concyclic intersections

MMO 295 Moscow MO 1955 convex domains on plane contains straight line

MMO 294 Moscow MO 1955 square table witn no 1-49, symmetric

MMO 293 Moscow MO 1955 locus, equal segments on sides of ABCD

MMO 292 Moscow MO 1955 2^n = 10a + b , 6 \ ab

MMO 291 Moscow MO 1955 rectangles cut into 13 equal squares

MMO 290 Moscow MO 1955 n^2 + n + 1 divisible by 1955

MMO 289 Moscow MO 1955 locus, related to equilateral and AE=CD

MMO 288 Moscow MO 1955 area wanted, circle bisects segment, right triangle

MMO 287 Moscow MO 1955 square table with numbers 1 to 49, sum

1955 Moscow Mathematical Olympiad

Subcontests

MMO 313- Moscow MO 1955 2 intersecting lines and a transfromation

MMO 319 Moscow MO 1955 triangles inscribed in triangles, equal ratio

MMO 318 Moscow MO 1955 max no of triples from 1955 points, each 2 1 common

MMO 317 Moscow MO 1955 source of light on right circular cone

MMO 316 Moscow MO 1955 p/q root of integer f(x) =&gt; (p - kq) /f(k)

MMO 315 Moscow MO 1955 5 men play dominoes, partner, opponent

MMO 314 Moscow MO 1955 x^n - a_1x^{n-1} -...-a_o=0, a_i&gt;=0

MMO 313 Moscow MO 1955 system of segments on line, arithm. progression

MMO 312 Moscow MO 1955 equal rations lead to parallels, &Delta; inscribed in &Delta;

MMO 311 Moscow MO 1955 [a], .., [Na] and [1/a],...[N/a], are distinct

MMO 310 Moscow MO 1955 given 6-variable inequality, triangle sides exist?

MMO 309 Moscow MO 1955 numbering sides of n-gon

MMO 308 Moscow MO 1955 3 tangent circlee, inner tangent bisects arc

MMO 307 Moscow MO 1955 ax^2+bx+c= square for any integer x

MMO 306 Moscow MO 1955 cut a rectangle into 18 rectangles

MMO 305 Moscow MO 1955 25 in a chess tournament

MMO 304 Moscow MO 1955 triangle of excenters is acute

MMO 303 Moscow MO 1955 ax^2+bx+c = 4-th power of integer x

MMO 302 Moscow MO 1955 diophantine x^3 - 2y^3 - 4z^3 = 0

MMO 301 Moscow MO 1955 &lt;OAB, &lt;OBA, &lt;OBC, &lt;OCB,&lt;OAC,&lt;OCA acute, trihedral

MMO 300 Moscow MO 1955 AC - DA &gt; 1, BC - BD &gt; 1, EC - ED &gt; 1

MMO 299 Moscow MO 1955 increasing arithmetic progression of primes

MMO 298 Moscow MO 1955 system x^3 + y^3 = 1, x^4 + y^4 = 1

MMO 297 Moscow MO 1955 locus of midpoints of segments on 2 circles

MMO 296 Moscow MO 1955 cyclic ABCD, 4 circles, concyclic intersections

MMO 295 Moscow MO 1955 convex domains on plane contains straight line

MMO 294 Moscow MO 1955 square table witn no 1-49, symmetric

MMO 293 Moscow MO 1955 locus, equal segments on sides of ABCD

MMO 292 Moscow MO 1955 2^n = 10a + b , 6 \ ab

MMO 291 Moscow MO 1955 rectangles cut into 13 equal squares

MMO 290 Moscow MO 1955 n^2 + n + 1 divisible by 1955

MMO 289 Moscow MO 1955 locus, related to equilateral and AE=CD

MMO 288 Moscow MO 1955 area wanted, circle bisects segment, right triangle

MMO 287 Moscow MO 1955 square table with numbers 1 to 49, sum

MMO 316 Moscow MO 1955 p/q root of integer f(x) => (p - kq) /f(k)

MMO 314 Moscow MO 1955 x^n - a_1x^{n-1} -...-a_o=0, a_i>=0

MMO 312 Moscow MO 1955 equal rations lead to parallels, Δ inscribed in Δ

MMO 301 Moscow MO 1955 <OAB, <OBA, <OBC, <OCB,<OAC,<OCA acute, trihedral

MMO 300 Moscow MO 1955 AC - DA > 1, BC - BD > 1, EC - ED > 1