MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan National Olympiad
1993 Taiwan National Olympiad
1993 Taiwan National Olympiad
Part of
Taiwan National Olympiad
Subcontests
(6)
4
1
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lattice point
In the Cartesian plane, let
C
C
C
be a unit circle with center at origin
O
O
O
. For any point
Q
Q
Q
in the plane distinct from
O
O
O
, define
Q
′
Q'
Q
′
to be the intersection of the ray
O
Q
OQ
OQ
and the circle
C
C
C
. Prove that for any
P
∈
C
P\in C
P
∈
C
and any
k
∈
N
k\in\mathbb{N}
k
∈
N
there exists a lattice point
Q
(
x
,
y
)
Q(x,y)
Q
(
x
,
y
)
with
∣
x
∣
=
k
|x|=k
∣
x
∣
=
k
or
∣
y
∣
=
k
|y|=k
∣
y
∣
=
k
such that
P
Q
′
<
1
2
k
PQ'<\frac{1}{2k}
P
Q
′
<
2
k
1
.
2
1
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exists a cricle
Let
E
E
E
and
F
F
F
are distinct points on the diagonal
A
C
AC
A
C
of a parallelogram
A
B
C
D
ABCD
A
BC
D
. Prove that , if there exists a cricle through
E
,
F
E,F
E
,
F
tangent to rays
B
A
,
B
C
BA,BC
B
A
,
BC
then there also exists a cricle through
E
,
F
E,F
E
,
F
tangent to rays
D
A
,
D
C
DA,DC
D
A
,
D
C
.
6
1
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\sum_{k=1}^n\frac{1}{\sin{k}\sin{k+1}}
Let
m
m
m
be equal to
1
1
1
or
2
2
2
and
n
<
10799
n<10799
n
<
10799
be a positive integer. Determine all such
n
n
n
for which
∑
k
=
1
n
1
sin
k
sin
(
k
+
1
)
=
m
sin
n
sin
2
1
\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}
∑
k
=
1
n
s
i
n
k
s
i
n
(
k
+
1
)
1
=
m
s
i
n
2
1
s
i
n
n
.
5
1
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$A=\{a_1,a_2,...,a_{12}\}
Assume
A
=
{
a
1
,
a
2
,
.
.
.
,
a
12
}
A=\{a_{1},a_{2},...,a_{12}\}
A
=
{
a
1
,
a
2
,
...
,
a
12
}
is a set of positive integers such that for each positive integer
n
≤
2500
n \leq 2500
n
≤
2500
there is a subset
S
S
S
of
A
A
A
whose sum of elements is
n
n
n
. If
a
1
<
a
2
<
.
.
.
<
a
12
a_{1}<a_{2}<...<a_{12}
a
1
<
a
2
<
...
<
a
12
, what is the smallest possible value of
a
1
a_{1}
a
1
?
1
1
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$a_n=[n+\sqrt{2}+\frac{1}{2}]$
A sequence
(
a
n
)
(a_{n})
(
a
n
)
of positive integers is given by
a
n
=
[
n
+
n
+
1
2
]
a_{n}=[n+\sqrt{n}+\frac{1}{2}]
a
n
=
[
n
+
n
+
2
1
]
. Find all of positive integers which belong to the sequence.
3
1
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Find all $x,y,z\in\mathbb{N}_0$
Find all
x
,
y
,
z
∈
N
0
x,y,z\in\mathbb{N}_{0}
x
,
y
,
z
∈
N
0
such that 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}. Alternative formulation: Solve the equation 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z} in nonnegative integers
x
x
x
,
y
y
y
,
z
z
z
.