MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1949 Moscow Mathematical Olympiad
1949 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(17)
172
1
Hide problems
MMO 172 Moscow MO 1949 8 non-overlapping squares intersections
Two squares are said to be juxtaposed if their intersection is a point or a segment. Prove that it is impossible to juxtapose to a square more than eight non-overlapping squares of the same size.
171
1
Hide problems
MMO 171 Moscow MO 1949 number of the form 2^n , combination digits
* Prove that a number of the form
2
n
2^n
2
n
for a positive integer
n
n
n
may begin with any given combination of digits.
170
1
Hide problems
MMO 170 Moscow MO 1949 max area central symmetric polygon in triangle
What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?
169
1
Hide problems
MMO 169 Moscow MO 1949 convex polyhedron construction
Construct a convex polyhedron of equal “bricks” shown in Figure. https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif
168
1
Hide problems
MMO 168 Moscow MO 1949 sum of integers divisible by 100
Prove that some (or one) of any
100
100
100
integers can always be chosen so that the sum of the chosen integers is divisible by
100
100
100
.
167
1
Hide problems
MMO 167 Moscow MO 1949 centroids coincide in a hexagon
The midpoints of alternative sides of a convex hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.
166
1
Hide problems
MMO 166 Moscow MO 1949 13 weights of integer mass
Consider
13
13
13
weights of integer mass (in grams). It is known that any
6
6
6
of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass.
165
1
Hide problems
MMO 165 Moscow MO 1949 locus, 2 triangles, parallelogram related
Consider two triangles,
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
, and any point
O
O
O
. We take any point
X
X
X
in
△
A
B
C
\vartriangle ABC
△
A
BC
and any point
Y
Y
Y
in
△
D
E
F
\vartriangle DEF
△
D
EF
and draw a parallelogram
O
X
Y
Z
OXY Z
OX
Y
Z
. Prove that the locus of all possible points
Z
Z
Z
form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.
164
1
Hide problems
MMO 164 Moscow MO 1949 12 points on a circle, 4 checkers
There are
12
12
12
points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?
163
1
Hide problems
MMO 163 Moscow MO 1949 hexagon criterion to be inscribed
Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.
162
1
Hide problems
MMO 162 Moscow MO 1949 foursomes of 4n a geometric progression
Given a set of
4
n
4n
4
n
positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least
4
4
4
numbers of the set are identical.
161
1
Hide problems
MMO 161 Moscow MO 1949 x^2+2ax+1/16=-a+\sqrt{a^2 +x-1/16}
Find the real roots of the equation
x
2
+
2
a
x
+
1
16
=
−
a
+
a
2
+
x
−
1
16
x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }
x
2
+
2
a
x
+
16
1
=
−
a
+
a
2
+
x
−
16
1
,
(
0
<
a
<
1
4
)
\left(0 < a < \frac14 \right)
(
0
<
a
<
4
1
)
.
160
1
Hide problems
MMO 160 Moscow MO 1949 circumcircle, incenter, excenter related
Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.
159
1
Hide problems
MMO 159 Moscow MO 1949 disc of radius 1/4 cover broken line perimeter 1
Consider a closed broken line of perimeter
1
1
1
on a plane. Prove that a disc of radius
1
4
\frac14
4
1
can cover this line.
158
1
Hide problems
MMO 158 Moscow MO 1949 diophantine x^2 + y^2 + z^2 + u^2 = 2xyzu
a) Prove that
x
2
+
y
2
+
z
2
=
2
x
y
z
x^2 + y^2 + z^2 = 2xyz
x
2
+
y
2
+
z
2
=
2
x
yz
for integer
x
,
y
,
z
x, y, z
x
,
y
,
z
only if
x
=
y
=
z
=
0
x = y = z = 0
x
=
y
=
z
=
0
.b) Find integers
x
,
y
,
z
,
u
x, y, z, u
x
,
y
,
z
,
u
such that
x
2
+
y
2
+
z
2
+
u
2
=
2
x
y
z
u
x^2 + y^2 + z^2 + u^2 = 2xyzu
x
2
+
y
2
+
z
2
+
u
2
=
2
x
yz
u
.
157
1
Hide problems
MMO 157 Moscow MO 1949 axes of symmetry
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
156
1
Hide problems
MMO 156 Moscow MO 1949 27 195^8 - 10 887^8 + 10 152^8 divisible by
Prove that
2719
5
8
−
1088
7
8
+
1015
2
8
27 195^8 - 10 887^8 + 10 152^8
2719
5
8
−
1088
7
8
+
1015
2
8
is divisible by
26460
26 460
26460
.