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Vojtěch Jarník IMC
2014 VJIMC
2014 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
parabolas and tangency, points conparabolic
Let
P
1
,
P
2
,
P
3
,
P
4
P_1,P_2,P_3,P_4
P
1
,
P
2
,
P
3
,
P
4
be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that
P
1
P_1
P
1
is tangent to
P
2
P_2
P
2
at the point
q
2
,
P
2
q_2,P_2
q
2
,
P
2
is tangent to
P
3
P_3
P
3
at the point
q
3
,
P
3
q_3,P_3
q
3
,
P
3
is tangent to
P
4
P_4
P
4
at the point
q
4
q_4
q
4
, and
P
4
P_4
P
4
is tangent to
P
1
P_1
P
1
at the point
q
1
q_1
q
1
. Assume that all the points
q
1
,
q
2
,
q
3
,
q
4
q_1,q_2,q_3,q_4
q
1
,
q
2
,
q
3
,
q
4
have distinct
x
x
x
-coordinates. Prove that
q
1
,
q
2
,
q
3
,
q
4
q_1,q_2,q_3,q_4
q
1
,
q
2
,
q
3
,
q
4
lie on a graph of an at most quadratic polynomial.
double integral inequality
Let
0
<
a
<
b
0<a<b
0
<
a
<
b
and let
f
:
[
a
,
b
]
→
R
f:[a,b]\to\mathbb R
f
:
[
a
,
b
]
→
R
be a continuous function with
∫
a
b
f
(
t
)
d
t
=
0
\int^b_af(t)dt=0
∫
a
b
f
(
t
)
d
t
=
0
. Show that
∫
a
b
∫
a
b
f
(
x
)
f
(
y
)
ln
(
x
+
y
)
d
x
d
y
≤
0.
\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.
∫
a
b
∫
a
b
f
(
x
)
f
(
y
)
ln
(
x
+
y
)
d
x
d
y
≤
0.
Problem 3
2
Hide problems
another tanh inequality
Let
n
≥
2
n\ge2
n
≥
2
be an integer and let
x
>
0
x>0
x
>
0
be a real number. Prove that
(
1
−
tanh
x
)
n
+
tanh
(
n
x
)
<
1.
\left(1-\sqrt{\tanh x}\right)^n+\sqrt{\tanh(nx)}<1.
(
1
−
tanh
x
)
n
+
tanh
(
n
x
)
<
1.
a binomial sum, alternating
Let
k
k
k
be a positive even integer. Show that
∑
n
=
0
k
/
2
(
−
1
)
n
(
k
+
2
n
)
(
2
(
k
−
n
)
+
1
k
+
1
)
=
(
k
+
1
)
(
k
+
2
)
2
.
\sum_{n=0}^{k/2}(-1)^n\binom{k+2}n\binom{2(k-n)+1}{k+1}=\frac{(k+1)(k+2)}2.
n
=
0
∑
k
/2
(
−
1
)
n
(
n
k
+
2
)
(
k
+
1
2
(
k
−
n
)
+
1
)
=
2
(
k
+
1
)
(
k
+
2
)
.
Problem 2
2
Hide problems
x+y=z in Fp * subgroup, 6|order
Let
p
p
p
be a prime number and let
A
A
A
be a subgroup of the multiplicative group
F
p
∗
\mathbb F^*_p
F
p
∗
of the finite field
F
p
\mathbb F_p
F
p
with
p
p
p
elements. Prove that if the order of
A
A
A
is a multiple of
6
6
6
, then there exist
x
,
y
,
z
∈
A
x,y,z\in A
x
,
y
,
z
∈
A
satisfying
x
+
y
=
z
x+y=z
x
+
y
=
z
.
faro shuffle 8 times =Id (VJIMC 2014 1.2)
We have a deck of
2
n
2n
2
n
cards. Each shuffling changes the order from
a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
n
a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n
a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
n
to
a
1
,
b
1
,
a
2
,
b
2
,
…
,
a
n
,
b
n
a_1,b_1,a_2,b_2,\ldots,a_n,b_n
a
1
,
b
1
,
a
2
,
b
2
,
…
,
a
n
,
b
n
. Determine all even numbers
2
n
2n
2
n
such that after shuffling the deck
8
8
8
times the original order is restored.
Problem 1
2
Hide problems
complex # equation (VJIMC 2014 1.1)
Find all complex numbers
z
z
z
such that
∣
z
3
+
2
−
2
i
∣
+
z
z
‾
∣
z
∣
=
2
2
.
|z^3+2-2i|+z\overline z|z|=2\sqrt2.
∣
z
3
+
2
−
2
i
∣
+
z
z
∣
z
∣
=
2
2
.
limit(∞) of f(x)+f'(x)/x given, find lim(∞) f(x)
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb R
f
:
(
0
,
∞
)
→
R
be a differentiable function. Assume that
lim
x
→
∞
(
f
(
x
)
+
f
′
(
x
)
x
)
=
0.
\lim_{x\to\infty}\left(f(x)+\frac{f'(x)}x\right)=0.
x
→
∞
lim
(
f
(
x
)
+
x
f
′
(
x
)
)
=
0.
Prove that
lim
x
→
∞
f
(
x
)
=
0.
\lim_{x\to\infty}f(x)=0.
x
→
∞
lim
f
(
x
)
=
0.