MathDB

1

MMO 207 Moscow MO 1951 bus route, 14 stops, 25 passengers

14

stops (counting the first and the last). A bus cannot carry more than

25

passengers. We assume that a passenger takes a bus from

A

to

B

if (s)he enters the bus at

A

and gets off at

B

. Prove that for any bus route:
a) there are

8

distinct stops

A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4

such that no passenger rides from

A_k

to

B_k

for all

k = 1, 2, 3, 4

(#)
b) there might not exist

10

distinct stops

A_1, B_1, . . . , A_5, B_5

with property (#).

206

1

MMO 206 Moscow MO 1951 find a closed planar curve without self-intersections

Consider a curve with the following property:
inside the curve one can move an inscribed equilateral triangle so that each vertex of the triangle moves along the curve and draws the whole curve.
Clearly, every circle possesses the property. Find a closed planar curve without self-intersections, that has the property but is not a circle.

205

1

MMO 205 Moscow MO 1951 max area reg. tetrahedron projection

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

204

1

MMO 204 Moscow MO 1951 sum of reciprocals < 2, lcm>1951

* Given several numbers each of which is less than

1951

and the least common multiple of any two of which is greater than

1951

. Prove that the sum of their reciprocals is less than

2

.

203

1

MMO 203 Moscow MO 1951 sphere inscribed in n-angled pyramid.

A sphere is inscribed in an

n

-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then (1) all tangent points of these faces to the sphere would coincide with one point,

H

, and (2) the vertices of the faces would lie on a circle centered at

H

.

202

1

MMO 202 Moscow MO 1951 (x^{1951}-1):(x^4+x^3+2x^2+x+1)

Dividing

x^{1951} - 1

by

P(x) = x^4 + x^3 + 2x^2 + x + 1

one gets a quotient and a remainder. Find the coefficient of

x^{14}

in the quotient.

201

1

MMO 201 Moscow MO 1951 20 problems to 20 students

To prepare for an Olympiad

20

students went to a coach. The coach gave them

20

problems and it turned out that
(a) each of the students solved two problems and (b) each problem was solved by twostudents.
Prove that it is possible to organize the coaching so that each student would discuss one of the problems that (s)he had solved, and so that all problems would be discussed.

200

1

MMO 200 Moscow MO 1951 central projection of a triangle

What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)

199

1

MMO 199 Moscow MO 1951 1^3 + 2^3 +...+ n^3 a perfect square

Prove that the sum

1^3 + 2^3 +...+ n^3

is a perfect square for all

n

.

198

1

MMO 198 Moscow MO 1951 ruler and protractor construction

* On a plane, given points

A, B, C

and angles

\angle D, \angle E, \angle F

each less than

180^o

and the sum equal to

360^o

, construct with the help of ruler and protractor a point

O

such that

\angle AOB = \angle D, \angle BOC = \angle E

and

\angle COA = \angle F.

197

1

MMO 197 Moscow MO 1951 10 ... 0 5 0...01 not a perfect cube

Prove that the number

1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1

is not the cube of any integer.

196

1

MMO 196 Moscow MO 1951 V = I R, 3 equidistant pallel lines, ruler

Given three equidistant parallel lines. Express by points of the corresponding lines the values of the resistance, voltage and current in a conductor so as to obtain the voltage

V = I \cdot R

by connecting with a ruler the points denoting the resistance

R

and the current

I

. (Each point of each scale denotes only one number).
[hide=similar wording]Three parallel straight lines are given at equal distances from each other. How to depict by points of the corresponding straight lines the values of resistance, voltage and the current in the conductor, so that, applying a ruler to to points depicting the values of resistance R and values of current I, obtain on the voltage scale a point depicting the value of voltage V = I R (point each scale represents one and only one number).

195

1

MMO 195 Moscow MO 1951 concentric circles and polygon, pyramid

We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?

194

1

MMO 194 Moscow MO 1951 largest area polygon, side a, sum of angles

One side of a convex polygon is equal to

a

, the sum of exterior angles at the vertices not adjacent to this side are equal to

120^o

. Among such polygons, find the polygon of the largest area.

193

1

MMO 193 Moscow MO 1951 first 3 figures after the decimal point

Prove that the first 3 digits after the decimal point in the decimal expression of the number

\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}

are

239

.

192

1

MMO 192 Moscow MO 1951 chain of 150 links each weighing 1g

a) Given a chain of

60

links each weighing

1

g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights

1

g,

2

g, . . . ,

59

g,

60

g? A broken link also weighs

1

g.
b) Given a chain of

150

links each weighing

1

g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights

1

g,

2

g, . . . ,

149

g,

150

g? A broken link also weighs

1

g.

191

1

MMO 191 Moscow MO 1951 isosceles trapezoid ABCD, P, construct quadril.

Given an isosceles trapezoid

ABCD

and a point

P

. Prove that a quadrilateral can be constructed from segments

PA, PB, PC, PD

.
Note: It is allowed that the vertices of a quadrilateral lie not only not only on the sides of the trapezoid, but also on their extensions.

190

1

MMO 190 Moscow MO 1951 compare numbers

Which number is greater:

\frac{2.00 000 000 004}{(1.00 000 000 004)^2 + 2.00 000 000 004}

or

\frac{2.00 000 000 002}{(1.00 000 000 002)^2 + 2.00 000 000 002}

?

189

1

MMO 189 Moscow MO 1951 ABCD,A'B'C'D' with equal sides, unequal angles

Let

ABCD

and

A'B'C'D'

be two convex quadrilaterals whose corresponding sides are equal, i.e.,

AB = A'B', BC = B'C'

, etc. Prove that if

\angle A > \angle A'

, then

\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'

.

188

1

MMO 188 Moscow MO 1951 x^{12} - x^9 + x^4 - x + 1 > 0

Prove that

x^{12} - x^9 + x^4 - x + 1 > 0

for all

x

.

1951 Moscow Mathematical Olympiad

Subcontests

MMO 207 Moscow MO 1951 bus route, 14 stops, 25 passengers

MMO 206 Moscow MO 1951 find a closed planar curve without self-intersections

MMO 205 Moscow MO 1951 max area reg. tetrahedron projection

MMO 204 Moscow MO 1951 sum of reciprocals < 2, lcm>1951

MMO 203 Moscow MO 1951 sphere inscribed in n-angled pyramid.

MMO 202 Moscow MO 1951 (x^{1951}-1):(x^4+x^3+2x^2+x+1)

MMO 201 Moscow MO 1951 20 problems to 20 students

MMO 200 Moscow MO 1951 central projection of a triangle

MMO 199 Moscow MO 1951 1^3 + 2^3 +...+ n^3 a perfect square

MMO 198 Moscow MO 1951 ruler and protractor construction

MMO 197 Moscow MO 1951 10 ... 0 5 0...01 not a perfect cube

MMO 196 Moscow MO 1951 V = I R, 3 equidistant pallel lines, ruler

MMO 195 Moscow MO 1951 concentric circles and polygon, pyramid

MMO 194 Moscow MO 1951 largest area polygon, side a, sum of angles

MMO 193 Moscow MO 1951 first 3 figures after the decimal point

MMO 192 Moscow MO 1951 chain of 150 links each weighing 1g

MMO 191 Moscow MO 1951 isosceles trapezoid ABCD, P, construct quadril.

MMO 190 Moscow MO 1951 compare numbers

MMO 189 Moscow MO 1951 ABCD,A'B'C'D' with equal sides, unequal angles

MMO 188 Moscow MO 1951 x^{12} - x^9 + x^4 - x + 1 > 0

1951 Moscow Mathematical Olympiad

Subcontests

MMO 207 Moscow MO 1951 bus route, 14 stops, 25 passengers

MMO 206 Moscow MO 1951 find a closed planar curve without self-intersections

MMO 205 Moscow MO 1951 max area reg. tetrahedron projection

MMO 204 Moscow MO 1951 sum of reciprocals &lt; 2, lcm&gt;1951

MMO 203 Moscow MO 1951 sphere inscribed in n-angled pyramid.

MMO 202 Moscow MO 1951 (x^{1951}-1):(x^4+x^3+2x^2+x+1)

MMO 201 Moscow MO 1951 20 problems to 20 students

MMO 200 Moscow MO 1951 central projection of a triangle

MMO 199 Moscow MO 1951 1^3 + 2^3 +...+ n^3 a perfect square

MMO 198 Moscow MO 1951 ruler and protractor construction

MMO 197 Moscow MO 1951 10 ... 0 5 0...01 not a perfect cube

MMO 196 Moscow MO 1951 V = I R, 3 equidistant pallel lines, ruler

MMO 195 Moscow MO 1951 concentric circles and polygon, pyramid

MMO 194 Moscow MO 1951 largest area polygon, side a, sum of angles

MMO 193 Moscow MO 1951 first 3 figures after the decimal point

MMO 192 Moscow MO 1951 chain of 150 links each weighing 1g

MMO 191 Moscow MO 1951 isosceles trapezoid ABCD, P, construct quadril.

MMO 190 Moscow MO 1951 compare numbers

MMO 189 Moscow MO 1951 ABCD,A'B'C'D' with equal sides, unequal angles

MMO 188 Moscow MO 1951 x^{12} - x^9 + x^4 - x + 1 &gt; 0

MMO 204 Moscow MO 1951 sum of reciprocals < 2, lcm>1951

MMO 188 Moscow MO 1951 x^{12} - x^9 + x^4 - x + 1 > 0