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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2012 Finnish National High School Mathematics Competition
2012 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Collatz function applied 40 times
The Collatz's function is a mapping
f
:
Z
+
→
Z
+
f:\mathbb{Z}_+\to\mathbb{Z}_+
f
:
Z
+
→
Z
+
satisfying f(x)=\begin{cases} 3x+1,& \mbox{as }x\mbox{ is odd}\\ x/2, & \mbox{as }x\mbox{ is even.}\\ \end{cases} In addition, let us define the notation
f
1
=
f
f^1=f
f
1
=
f
and inductively
f
k
+
1
=
f
∘
f
k
,
f^{k+1}=f\circ f^k,
f
k
+
1
=
f
∘
f
k
,
or to say in another words,
f
k
(
x
)
=
f
(
…
(
f
⏟
k
times
(
x
)
…
)
.
f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).
f
k
(
x
)
=
k
times
f
(
…
(
f
(
x
)
…
)
.
Prove that there is an
x
∈
Z
+
x\in\mathbb{Z}_+
x
∈
Z
+
satisfying
f
40
(
x
)
>
2012
x
.
f^{40}(x)> 2012x.
f
40
(
x
)
>
2012
x
.
4
1
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Combinatorial Inequality
Let
k
,
n
∈
N
,
0
<
k
<
n
.
k,n\in\mathbb{N},0<k<n.
k
,
n
∈
N
,
0
<
k
<
n
.
Prove that
∑
j
=
1
k
(
n
j
)
=
(
n
1
)
+
(
n
2
)
+
…
+
(
n
k
)
≤
n
k
.
\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.
j
=
1
∑
k
(
j
n
)
=
(
1
n
)
+
(
2
n
)
+
…
+
(
k
n
)
≤
n
k
.
3
1
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$(k-1)^2$ divides $k^{k-1}-1$
Prove that for all integers
k
≥
2
,
k\geq 2,
k
≥
2
,
the number
k
k
−
1
−
1
k^{k-1}-1
k
k
−
1
−
1
is divisible by
(
k
−
1
)
2
.
(k-1)^2.
(
k
−
1
)
2
.
2
1
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Given two equal quantities, show they are both x+y+z
Let
x
≠
1
,
y
≠
1
x\ne 1,y\ne 1
x
=
1
,
y
=
1
and
x
≠
y
.
x\ne y.
x
=
y
.
Show that if
y
z
−
x
2
1
−
x
=
z
x
−
y
2
1
−
y
,
\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},
1
−
x
yz
−
x
2
=
1
−
y
z
x
−
y
2
,
then
y
z
−
x
2
1
−
x
=
z
x
−
y
2
1
−
y
=
x
+
y
+
z
.
\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.
1
−
x
yz
−
x
2
=
1
−
y
z
x
−
y
2
=
x
+
y
+
z
.
1
1
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Secant splits a circle
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is
5
:
9
5:9
5
:
9
. Compute the ratio of the secant line versus circle radius.