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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1987 China Team Selection Test
1987 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
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China TST 1987 sequence challenge
Let
r
1
=
2
r_1=2
r
1
=
2
and
r
n
=
∏
k
=
1
n
−
1
r
i
+
1
r_n = \prod^{n-1}_{k=1} r_i + 1
r
n
=
∏
k
=
1
n
−
1
r
i
+
1
,
n
≥
2.
n \geq 2.
n
≥
2.
Prove that among all sets of positive integers such that
∑
k
=
1
n
1
a
i
<
1
,
\sum^{n}_{k=1} \frac{1}{a_i} < 1,
∑
k
=
1
n
a
i
1
<
1
,
the partial sequences
r
1
,
r
2
,
.
.
.
,
r
n
r_1,r_2, ... , r_n
r
1
,
r
2
,
...
,
r
n
are the one that gets nearer to 1.
two triangles with a common edge [variation on Turan]
Let
G
G
G
be a simple graph with
2
⋅
n
2 \cdot n
2
⋅
n
vertices and
n
2
+
1
n^{2}+1
n
2
+
1
edges. Show that this graph
G
G
G
contains a
K
4
−
one edge
K_{4}-\text{one edge}
K
4
−
one edge
, that is, two triangles with a common edge.
2
2
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closed recticular polygon
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
x^3+y^3+z^3=nx^2y^2z^2
Find all positive integer
n
n
n
such that the equation
x
3
+
y
3
+
z
3
=
n
⋅
x
2
⋅
y
2
⋅
z
2
x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2
x
3
+
y
3
+
z
3
=
n
⋅
x
2
⋅
y
2
⋅
z
2
has positive integer solutions.
1
2
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China TST 1987 smallest integer from five sets
a.) For all positive integer
k
k
k
find the smallest positive integer
f
(
k
)
f(k)
f
(
k
)
such that
5
5
5
sets
s
1
,
s
2
,
…
,
s
5
s_1,s_2, \ldots , s_5
s
1
,
s
2
,
…
,
s
5
exist satisfying: i. each has
k
k
k
elements; ii.
s
i
s_i
s
i
and
s
i
+
1
s_{i+1}
s
i
+
1
are disjoint for
i
=
1
,
2
,
.
.
.
,
5
i=1,2,...,5
i
=
1
,
2
,
...
,
5
(
s
6
=
s
1
s_6=s_1
s
6
=
s
1
) iii. the union of the
5
5
5
sets has exactly
f
(
k
)
f(k)
f
(
k
)
elements.b.) Generalisation: Consider
n
≥
3
n \geq 3
n
≥
3
sets instead of
5
5
5
.
construct a rectangle A inside it with maximum area
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle
A
A
A
inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.