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Undergraduate contests
VTRMC
2017 VTRMC
2017 VTRMC
Part of
VTRMC
Subcontests
(7)
7
1
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2017 VTRMC #7
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of nonnegative integers for which
m
2
+
2
⋅
3
n
=
m
(
2
n
+
1
−
1
)
m ^ { 2 } + 2 \cdot 3 ^ { n } = m \left( 2 ^ { n + 1 } - 1 \right)
m
2
+
2
⋅
3
n
=
m
(
2
n
+
1
−
1
)
.
6
1
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2017 VTRMC #6
Let
f
(
x
)
∈
Z
[
x
]
f ( x ) \in \mathbb { Z } [ x ]
f
(
x
)
∈
Z
[
x
]
be a polynomial with integer coefficients such that
f
(
1
)
=
−
1
,
f
(
4
)
=
2
f ( 1 ) = - 1 , f ( 4 ) = 2
f
(
1
)
=
−
1
,
f
(
4
)
=
2
and
f
(
8
)
=
34
f ( 8 ) = 34
f
(
8
)
=
34
. Suppose
n
∈
Z
n\in\mathbb{Z}
n
∈
Z
is an integer such that
f
(
n
)
=
n
2
−
4
n
−
18
f ( n ) = n ^ { 2 } - 4 n - 18
f
(
n
)
=
n
2
−
4
n
−
18
. Determine all possible values for
n
n
n
.
5
1
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2017 VTRMC #5
Let
f
(
x
,
y
)
=
(
x
+
y
)
/
2
,
g
(
x
,
y
)
=
x
y
,
h
(
x
,
y
)
=
2
x
y
/
(
x
+
y
)
f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y )
f
(
x
,
y
)
=
(
x
+
y
)
/2
,
g
(
x
,
y
)
=
x
y
,
h
(
x
,
y
)
=
2
x
y
/
(
x
+
y
)
, and let
S
=
{
(
a
,
b
)
∈
N
×
N
∣
a
≠
b
and
f
(
a
,
b
)
,
g
(
a
,
b
)
,
h
(
a
,
b
)
∈
N
}
S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \}
S
=
{(
a
,
b
)
∈
N
×
N
∣
a
=
b
and
f
(
a
,
b
)
,
g
(
a
,
b
)
,
h
(
a
,
b
)
∈
N
}
where
N
\mathbb{N}
N
denotes the positive integers. Find the minimum of
f
f
f
over
S
S
S
.
4
1
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2017 VTRMC #4
Let
P
P
P
be an interior point of a triangle of area
T
T
T
. Through the point
P
P
P
, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let
a
a
a
,
b
b
b
and
c
c
c
be the areas of the three triangles. Prove that
T
=
a
+
b
+
c
\sqrt { T } = \sqrt { a } + \sqrt { b } + \sqrt { c }
T
=
a
+
b
+
c
.
3
1
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2017 VTRMC #3
Let
A
B
C
ABC
A
BC
be a triangle and let
P
P
P
be a point in its interior. Suppose
∠
B
A
P
=
1
0
∘
,
∠
A
B
P
=
2
0
∘
,
∠
P
C
A
=
3
0
∘
\angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ }
∠
B
A
P
=
1
0
∘
,
∠
A
BP
=
2
0
∘
,
∠
PC
A
=
3
0
∘
and
∠
P
A
C
=
4
0
∘
\angle P A C = 40 ^ { \circ }
∠
P
A
C
=
4
0
∘
. Find
∠
P
B
C
\angle P B C
∠
PBC
.
2
1
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2017 VTRMC #2
Evaluate
∫
0
a
d
x
/
(
1
+
cos
x
+
sin
x
)
\int _ { 0 } ^ { a } d x / ( 1 + \cos x + \sin x )
∫
0
a
d
x
/
(
1
+
cos
x
+
sin
x
)
for
−
π
/
2
<
a
<
π
- \pi / 2 < a < \pi
−
π
/2
<
a
<
π
. Use your answer to show that
∫
0
π
/
2
d
x
/
(
1
+
cos
x
+
sin
x
)
=
ln
2
\int _ { 0 } ^ { \pi / 2 } d x / ( 1 + \cos x + \sin x ) = \ln 2
∫
0
π
/2
d
x
/
(
1
+
cos
x
+
sin
x
)
=
ln
2
.
1
1
Hide problems
VTRMC 2017 #1
Determine the number of real solutions to the equation
2
−
x
2
=
3
−
x
3
3
.
\sqrt{2 -x^2} = \sqrt[3]{3 -x^3}.
2
−
x
2
=
3
3
−
x
3
.