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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1988 Mexico National Olympiad
1988 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(8)
8
1
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volume of regular octahedron circumscribed about sphere of radius 1
Compute the volume of a regular octahedron circumscribed about a sphere of radius
1
1
1
.
7
1
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2 disjoint subsets of {1,2, ... ,m} with same sums of elements, max cardinal
Two disjoint subsets of the set
{
1
,
2
,
.
.
.
,
m
}
\{1,2, ... ,m\}
{
1
,
2
,
...
,
m
}
have the same sums of elements. Prove that each of the subsets
A
,
B
A,B
A
,
B
has less than
m
/
2
m / \sqrt2
m
/
2
elements.
6
1
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A,B,C on a circle, B,C fixed, A moving, find locus of incenters of ABC
Consider two fixed points
B
,
C
B,C
B
,
C
on a circle
w
w
w
. Find the locus of the incenters of all triangles
A
B
C
ABC
A
BC
when point
A
A
A
describes
w
w
w
.
5
1
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gcd of a^2+b^2-nab and a+b divides n+2 , when a,b are coprime positive
If
a
a
a
and
b
b
b
are coprime positive integers and
n
n
n
an integer, prove that the greatest common divisor of
a
2
+
b
2
−
n
a
b
a^2+b^2-nab
a
2
+
b
2
−
nab
and
a
+
b
a+b
a
+
b
divides
n
+
2
n+2
n
+
2
.
4
1
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no of ways of selecting 8 integers a_i such that 1<a_1<=...<=a_8<=8
In how many ways can one select eight integers
a
1
,
a
2
,
.
.
.
,
a
8
a_1,a_2, ... ,a_8
a
1
,
a
2
,
...
,
a
8
, not necesarily distinct, such that
1
≤
a
1
≤
.
.
.
≤
a
8
≤
8
1 \le a_1 \le ... \le a_8 \le 8
1
≤
a
1
≤
...
≤
a
8
≤
8
?
3
1
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triangle between common tangents of externally tangent circles in terms of r,R
Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.
2
1
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11a+2b is a multiple of 19 iff so is 18a+5b .
If
a
a
a
and
b
b
b
are positive integers, prove that
11
a
+
2
b
11a+2b
11
a
+
2
b
is a multiple of
19
19
19
if and only if so is
18
a
+
5
b
18a+5b
18
a
+
5
b
.
1
1
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arranging 7 white and 5 black balls in a line without 2 black neighbours
In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls?