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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1983 Czech and Slovak Olympiad III A
1983 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Finding a locus
Consider a circle
k
k
k
with center
S
S
S
and radius
r
r
r
. Denote
M
\mathsf M
M
the set of all triangles with incircle
k
k
k
such that the largest inner angle is twice bigger than the smallest one. For a triangle
T
∈
M
\mathcal T\in\mathsf M
T
∈
M
denote its vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
in way that
S
A
≥
S
B
≥
S
C
SA\ge SB\ge SC
S
A
≥
SB
≥
SC
. Find the locus of points
{
B
∣
T
∈
M
}
\{B\mid\mathcal T\in\mathsf M\}
{
B
∣
T
∈
M
}
.
5
1
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Inequality
Find all pair
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers satisfying
∣
x
y
−
2
∣
<
1
y
3
.
\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.
y
x
−
2
<
y
3
1
.
4
1
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Identity with binomial coefficients
Consider an arithmetic progression
a
0
,
…
,
a
n
a_0,\ldots,a_n
a
0
,
…
,
a
n
with
n
≥
2
n\ge2
n
≥
2
. Prove that
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
a
k
=
0.
\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.
k
=
0
∑
n
(
−
1
)
k
(
k
n
)
a
k
=
0.
3
1
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Rectangle on a chessboard
An
8
×
8
8\times 8
8
×
8
chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
2
1
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Inequality in a triangle
Given a triangle
A
B
C
ABC
A
BC
, prove that for every inner point
P
P
P
of the side
A
B
AB
A
B
the inequality
P
C
⋅
A
B
<
P
A
⋅
B
C
+
P
B
⋅
A
C
PC\cdot AB<PA\cdot BC+PB\cdot AC
PC
⋅
A
B
<
P
A
⋅
BC
+
PB
⋅
A
C
holds.
1
1
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Maximization of a sum of sines
Let
n
n
n
be a positive integer and
k
∈
[
0
,
n
]
k\in[0,n]
k
∈
[
0
,
n
]
be a fixed real constant. Find the maximum value of
∣
∑
i
=
1
n
sin
(
2
x
i
)
∣
\left|\sum_{i=1}^n\sin(2x_i)\right|
i
=
1
∑
n
sin
(
2
x
i
)
where
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
are real numbers satisfying
∑
i
=
1
n
sin
2
(
x
i
)
=
k
.
\sum_{i=1}^n\sin^2(x_i)=k.
i
=
1
∑
n
sin
2
(
x
i
)
=
k
.