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Problems
Contests
Undergraduate contests
VTRMC
2010 VTRMC
2010 VTRMC
Part of
VTRMC
Subcontests
(7)
Problem 5
1
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circle internally tangent to another
Let
A
,
B
A,B
A
,
B
be two circles in the plane with
B
B
B
inside
A
A
A
. Assume that
A
A
A
has radius
3
3
3
,
B
B
B
has radius
1
1
1
,
P
P
P
is a point on
A
A
A
,
Q
Q
Q
is a point on
B
B
B
, and
A
A
A
and
B
B
B
touch so that
P
P
P
and
Q
Q
Q
are the same point. Suppose that
A
A
A
is kept fixed and
B
B
B
is rolled once round the inside of
A
A
A
so that
Q
Q
Q
traces out a curve starting and finishing at
P
P
P
. What is the area enclosed by this curve? https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS84LzkwMDBjOTAwODk5M2QyM2IxMGUxZGE5OTI1NWU1ZDYwMDkyYTUwLnBuZw==&rn=VlRSTUMgMjAxMC5wbmc=
Problem 4
1
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prove side equation given angle equation
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
and corresponding angles
A
,
B
,
C
A,B,C
A
,
B
,
C
(so
a
=
B
C
a=BC
a
=
BC
and
A
=
∠
B
A
C
A=\angle BAC
A
=
∠
B
A
C
etc.). Suppose that
4
A
+
3
C
=
54
0
∘
4A+3C=540^\circ
4
A
+
3
C
=
54
0
∘
. Prove that
(
a
−
b
)
2
(
a
+
b
)
=
b
c
2
(a-b)^2(a+b)=bc^2
(
a
−
b
)
2
(
a
+
b
)
=
b
c
2
.
Problem 2
1
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75^75^75^ (2^100 times) mod 17
For
n
n
n
a positive integer, define
f
1
(
n
)
=
n
f_1(n)=n
f
1
(
n
)
=
n
and then for
i
i
i
a positive integer, define
f
i
+
1
(
n
)
=
f
i
(
n
)
f
i
(
n
)
f_{i+1}(n)=f_i(n)^{f_i(n)}
f
i
+
1
(
n
)
=
f
i
(
n
)
f
i
(
n
)
. Determine
f
100
(
75
)
(
m
o
d
17
)
f_{100}(75)\pmod{17}
f
100
(
75
)
(
mod
17
)
. Justify your answer.
Problem 7
1
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convergence of sequence sum with 1/(na_n^2)
Let
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
be a convergent series of positive terms (so
a
i
>
0
a_i>0
a
i
>
0
for all
i
i
i
) and set
b
n
=
1
n
a
n
2
b_n=\frac1{na_n^2}
b
n
=
n
a
n
2
1
for
n
≥
1
n\ge1
n
≥
1
. Prove that
∑
n
=
1
∞
n
b
1
+
b
2
+
…
+
b
n
\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}
∑
n
=
1
∞
b
1
+
b
2
+
…
+
b
n
n
is convergent.
Problem 6
1
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limit of sequence
Define a sequence by
a
1
=
1
,
a
2
=
1
2
a_1=1,a_2=\frac12
a
1
=
1
,
a
2
=
2
1
, and
a
n
+
2
=
a
n
+
1
−
a
n
a
n
+
1
2
a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2
a
n
+
2
=
a
n
+
1
−
2
a
n
a
n
+
1
for
n
n
n
a positive integer. Find
lim
n
→
∞
n
a
n
\lim_{n\to\infty}na_n
lim
n
→
∞
n
a
n
.
Problem 1
1
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determinant of matrix sum
Let
d
d
d
be a positive integer and let
A
A
A
be a
d
×
d
d\times d
d
×
d
matrix with integer entries. Suppose
I
+
A
+
A
2
+
…
+
A
100
=
0
I+A+A_2+\ldots+A_{100}=0
I
+
A
+
A
2
+
…
+
A
100
=
0
(where
I
I
I
denotes the identity
d
×
d
d\times d
d
×
d
matrix, and
0
0
0
denotes the zero matrix, which has all entries
0
0
0
). Determine the positive integers
n
≤
100
n\le100
n
≤
100
for which
A
n
+
A
n
+
1
+
…
+
A
100
A_n+A_{n+1}+\ldots+A_{100}
A
n
+
A
n
+
1
+
…
+
A
100
has determinant
±
1
\pm1
±
1
.
Problem 3
1
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equation -cubic
Solve in
R
R
R
the equation:
8
x
3
−
4
x
2
−
4
x
+
1
=
0
8x^3-4x^2-4x+1=0
8
x
3
−
4
x
2
−
4
x
+
1
=
0