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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1991 Mexico National Olympiad
1991 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
2
1
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(palindrome) if soldiers arrange in rows of 3,4,5, then last row contains 2,3,5
A company of
n
n
n
soldiers is such that (i)
n
n
n
is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of
3
,
4
3, 4
3
,
4
or
5
5
5
soldiers, then the last row contains
2
,
3
2, 3
2
,
3
and
5
5
5
soldiers, respectively. Find the smallest
n
n
n
satisfying these conditions and prove that there are infinitely many such numbers
n
n
n
.
6
1
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set Τ: Every two triangles in T have either two common vertices, or none.
Given an
n
n
n
-gon (
n
≥
4
n\ge 4
n
≥
4
), consider a set
T
T
T
of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that
T
T
T
contains at most
n
n
n
triangles.
5
1
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11 consecutive integers the sum of whose squares is a square.
The sum of squares of two consecutive integers can be a square, as in
3
2
+
4
2
=
5
2
3^2+4^2 =5^2
3
2
+
4
2
=
5
2
. Prove that the sum of squares of
m
m
m
consecutive integers cannot be a square for
m
=
3
m = 3
m
=
3
or
6
6
6
and find an example of
11
11
11
consecutive integers the sum of whose squares is a square.
4
1
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8 points concyclic
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a convex quarilateral
A
B
C
D
ABCD
A
BC
D
are orthogonal. Let
M
,
N
,
R
,
S
M,N,R,S
M
,
N
,
R
,
S
be the midpoints of the sides
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively, and let
W
,
X
,
Y
,
Z
W,X,Y,Z
W
,
X
,
Y
,
Z
be the projections of the points
M
,
N
,
R
M,N,R
M
,
N
,
R
and
S
S
S
on the lines
C
D
,
D
A
,
A
B
CD,DA,AB
C
D
,
D
A
,
A
B
and
B
C
BC
BC
, respectively. Prove that the points
M
,
N
,
R
,
S
,
W
,
X
,
Y
M,N,R,S,W,X,Y
M
,
N
,
R
,
S
,
W
,
X
,
Y
and
Z
Z
Z
lie on a circle.
3
1
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radius of the smallest sphere containing 4 equal balls, each tangent to other 3
Four balls of radius
1
1
1
are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?
1
1
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sum of all positive irreducible fractions <1 with denominator 1991
Evaluate the sum of all positive irreducible fractions less than
1
1
1
and having the denominator
1991
1991
1991
.