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Undergraduate contests
VTRMC
2022 VTRMC
2022 VTRMC
Part of
VTRMC
Subcontests
(6)
6
1
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Bounding the Derivative
Let
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a function whose second derivative is continuous. Suppose that
f
f
f
and
f
′
′
f''
f
′′
are bounded. Show that
f
′
f'
f
′
is also bounded.
5
1
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Eigenvalues of a Matrix
Let
A
A
A
be an invertible
n
×
n
n \times n
n
×
n
matrix with complex entries. Suppose that for each positive integer
m
m
m
, there exists a positive integer
k
m
k_m
k
m
and an
n
×
n
n \times n
n
×
n
invertible matrix
B
m
B_m
B
m
such that
A
k
m
m
=
B
m
A
B
m
−
1
A^{k_m m} = B_m A B_m ^{-1}
A
k
m
m
=
B
m
A
B
m
−
1
. Show that all eigenvalues of
A
A
A
are equal to
1
1
1
.
4
1
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Sum of Difference of Logs
Calculate the exact value of the series
∑
n
=
2
∞
log
(
n
3
+
1
)
−
log
(
n
3
−
1
)
\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)
∑
n
=
2
∞
lo
g
(
n
3
+
1
)
−
lo
g
(
n
3
−
1
)
and provide justification.
3
1
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Diophantine of a Single Base
Find all positive integers
a
,
b
,
c
,
d
,
a, b, c, d,
a
,
b
,
c
,
d
,
and
n
n
n
satisfying
n
a
+
n
b
+
n
c
=
n
d
n^a + n^b + n^c = n^d
n
a
+
n
b
+
n
c
=
n
d
and prove that these are the only such solutions.
2
1
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Focal Radii and Ellipse Tangent
Let
A
A
A
and
B
B
B
be the two foci of an ellipse and let
P
P
P
be a point on this ellipse. Prove that the focal radii of
P
P
P
(that is, the segments
A
P
‾
\overline{AP}
A
P
and
B
P
‾
\overline{BP}
BP
) form equal angles with the tangent to the ellipse at
P
P
P
.
1
1
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Sum of Consecutive Positive Integers
Give all possible representations of
2022
2022
2022
as a sum of at least two consecutive positive integers and prove that these are the only representations.