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Undergraduate contests
Vojtěch Jarník IMC
2012 VJIMC
2012 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
n^k starts/ends with same digit
Find all positive integers
n
n
n
for which there exists a positive integer
k
k
k
such that the decimal representation of
n
k
n^k
n
k
starts and ends with the same digit.
inequality in 7 variables, [1,4]
Let
a
,
b
,
c
,
x
,
y
,
z
,
t
a,b,c,x,y,z,t
a
,
b
,
c
,
x
,
y
,
z
,
t
be positive real numbers with
1
≤
x
,
y
,
z
≤
4
1\le x,y,z\le4
1
≤
x
,
y
,
z
≤
4
. Prove that
x
(
2
a
)
t
+
y
(
2
b
)
t
+
z
(
2
c
)
t
≥
y
+
z
−
x
(
b
+
c
)
t
+
z
+
x
−
y
(
c
+
a
)
t
+
x
+
y
−
z
(
a
+
b
)
t
.
\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.
(
2
a
)
t
x
+
(
2
b
)
t
y
+
(
2
c
)
t
z
≥
(
b
+
c
)
t
y
+
z
−
x
+
(
c
+
a
)
t
z
+
x
−
y
+
(
a
+
b
)
t
x
+
y
−
z
.
Problem 3
2
Hide problems
inequality, sum 1/(x+1)=1
Determine the smallest real number
C
C
C
such that the inequality
x
y
z
⋅
1
x
+
1
+
y
z
x
⋅
1
y
+
1
+
z
x
y
⋅
1
x
+
1
≤
C
\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C
yz
x
⋅
x
+
1
1
+
z
x
y
⋅
y
+
1
1
+
x
y
z
⋅
x
+
1
1
≤
C
holds for all positive real numbers
x
,
y
x,y
x
,
y
and
z
z
z
with
1
x
+
1
+
1
y
+
1
+
1
z
+
1
=
1
\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1
x
+
1
1
+
y
+
1
1
+
z
+
1
1
=
1
.
ring is commutative, either x^2=1 or x^n=0
Let
(
A
,
+
,
⋅
)
(A,+,\cdot)
(
A
,
+
,
⋅
)
be a ring with unity, having the following property: for all
x
∈
A
x\in A
x
∈
A
either
x
2
=
1
x^2=1
x
2
=
1
or
x
n
=
0
x^n=0
x
n
=
0
for some
n
∈
N
n\in\mathbb N
n
∈
N
. Show that
A
A
A
is a commutative ring.
Problem 2
2
Hide problems
A in M2(prime), A=B^2 and det(B)=p^2
Determine all
2
×
2
2\times2
2
×
2
integer matrices
A
A
A
having the following properties:
1.
1.
1.
the entries of
A
A
A
are (positive) prime numbers,
2.
2.
2.
there exists a
2
×
2
2\times2
2
×
2
integer matrix
B
B
B
such that
A
=
B
2
A=B^2
A
=
B
2
and the determinant of
B
B
B
is the square of a prime number.
eigenvalues of tridiagonal matrix
Let
M
M
M
be the (tridiagonal)
10
×
10
10\times10
10
×
10
matrix
M
=
(
−
1
3
0
⋯
⋯
⋯
0
3
2
−
1
0
⋮
0
−
1
2
−
1
⋱
⋮
⋮
0
−
1
2
⋱
0
⋮
⋮
⋱
⋱
⋱
−
1
0
⋮
0
−
1
2
−
1
0
⋯
⋯
⋯
0
−
1
2
)
M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}
M
=
−
1
3
0
⋮
⋮
⋮
0
3
2
−
1
0
⋯
0
−
1
2
−
1
⋱
⋯
⋯
0
−
1
2
⋱
0
⋯
⋯
⋱
⋱
⋱
−
1
0
⋯
0
−
1
2
−
1
0
⋮
⋮
⋮
0
−
1
2
Show that
M
M
M
has exactly nine positive real eigenvalues (counted with multiplicities).
Problem 1
2
Hide problems
|f'(x)|≠1, f has a fixed point and f(b)=1-b
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\to[0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
be a differentiable function such that
∣
f
′
(
x
)
∣
≠
1
|f'(x)|\ne1
∣
f
′
(
x
)
∣
=
1
for all
x
∈
[
0
,
1
]
x\in[0,1]
x
∈
[
0
,
1
]
. Prove that there exist unique
α
,
β
∈
[
0
,
1
]
\alpha,\beta\in[0,1]
α
,
β
∈
[
0
,
1
]
such that
f
(
α
)
=
α
f(\alpha)=\alpha
f
(
α
)
=
α
and
f
(
β
)
=
1
−
β
f(\beta)=1-\beta
f
(
β
)
=
1
−
β
.
given limsup, integral converges
Let
f
:
[
1
,
∞
)
→
(
0
,
∞
)
f:[1,\infty)\to(0,\infty)
f
:
[
1
,
∞
)
→
(
0
,
∞
)
be a non-increasing function such that
lim sup
n
→
∞
f
(
2
n
+
1
)
f
(
2
n
)
<
1
2
.
\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.
n
→
∞
lim
sup
f
(
2
n
)
f
(
2
n
+
1
)
<
2
1
.
Prove that
∫
1
∞
f
(
x
)
d
x
<
∞
.
\int^\infty_1f(x)\text dx<\infty.
∫
1
∞
f
(
x
)
d
x
<
∞.