MathDB
Problems
Contests
Undergraduate contests
VTRMC
2009 VTRMC
2009 VTRMC
Part of
VTRMC
Subcontests
(7)
Problem 6
1
Hide problems
n^4-7n^2+1 nonsquare
Let
n
n
n
be a nonzero integer. Prove that
n
4
−
7
n
2
+
1
n^4-7n^2+1
n
4
−
7
n
2
+
1
can never be a perfect square.
Problem 4
1
Hide problems
tangent circles, line through intersection
Two circles
α
,
β
\alpha,\beta
α
,
β
touch externally at the point
X
X
X
. Let
A
,
P
A,P
A
,
P
be two distinct points on
α
\alpha
α
different from
X
X
X
, and let
A
X
AX
A
X
and
P
X
PX
PX
meet
β
\beta
β
again in the points
B
B
B
and
Q
Q
Q
respectively. Prove that
A
P
AP
A
P
is parallel to
Q
B
QB
QB
.
Problem 2
1
Hide problems
digits of 40!
Given that
40
!
=
a
b
c
d
e
f
283247897734345611269596115894272
p
q
r
s
t
u
v
w
x
‾
40!=\overline{abcdef283247897734345611269596115894272pqrstuvwx}
40
!
=
ab
c
d
e
f
283247897734345611269596115894272
pq
rs
t
uv
w
x
, find
a
,
b
,
c
,
d
,
e
,
f
,
p
,
q
,
r
,
s
,
t
,
u
,
v
,
w
,
x
a,b,c,d,e,f,p,q,r,s,t,u,v,w,x
a
,
b
,
c
,
d
,
e
,
f
,
p
,
q
,
r
,
s
,
t
,
u
,
v
,
w
,
x
.
Problem 1
1
Hide problems
rate problem
A walker and a jogger travel along the same straight line in the same direction. The walker walks at one meter per second, while the jogger runs at two meters per second. The jogger starts one meter in front of the walker. A dog starts with the walker, and then runs back and forth between the walker and the jogger with constant speed of three meters per second. Let
f
(
n
)
f(n)
f
(
n
)
meters denote the total distance travelled by the dog when it has returned to the walker for the nth time (so
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
). Find a formula for
f
(
n
)
f(n)
f
(
n
)
.
Problem 7
1
Hide problems
existence of DE solution
Does there exist a twice differentiable function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that
f
′
(
x
)
=
f
(
x
+
1
)
−
f
(
x
)
f'(x)=f(x+1)-f(x)
f
′
(
x
)
=
f
(
x
+
1
)
−
f
(
x
)
for all
x
x
x
and
f
′
′
(
0
)
≠
0
f''(0)\ne0
f
′′
(
0
)
=
0
? Justify your answer.
Problem 5
1
Hide problems
3x3 matrices in C
Suppose
A
,
B
∈
M
3
(
C
)
A,B\in M_3(\mathbb C)
A
,
B
∈
M
3
(
C
)
,
B
≠
0
B\ne0
B
=
0
, and
A
B
=
0
AB=0
A
B
=
0
. Prove that there exists
D
∈
M
3
(
C
)
D\in M_3(\mathbb C)
D
∈
M
3
(
C
)
with
D
≠
0
D\ne0
D
=
0
such that
A
D
=
D
A
=
0
AD=DA=0
A
D
=
D
A
=
0
.
Problem 3
1
Hide problems
a double integral function
Define
f
(
x
)
=
∫
0
x
∫
0
x
e
u
2
v
2
d
u
d
v
f(x)=\int^x_0\int^x_0e^{u^2v^2}dudv
f
(
x
)
=
∫
0
x
∫
0
x
e
u
2
v
2
d
u
d
v
. Calculate
2
f
′
′
(
2
)
+
f
′
(
2
)
2f''(2)+f'(2)
2
f
′′
(
2
)
+
f
′
(
2
)
.