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Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2012 Kyrgyzstan National Olympiad
2012 Kyrgyzstan National Olympiad
Part of
Kyrgyzstan National Olympiad
Subcontests
(6)
6
1
Hide problems
1 through 50 on blackboard; replace any two with difference
The numbers
1
,
2
,
…
,
50
1, 2,\ldots, 50
1
,
2
,
…
,
50
are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?
5
1
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Prove that $a_k-22$ where $a_{k+2} = a_{k+1}a_k + 1$
The sequence of natural numbers is defined as follows: for any
k
≥
1
k\geq 1
k
≥
1
,
a
k
+
2
=
a
k
+
1
⋅
a
k
+
1
a_{k+2}= a_{k+1}\cdot a_k+1
a
k
+
2
=
a
k
+
1
⋅
a
k
+
1
. Prove that for
k
≥
9
k\geq 9
k
≥
9
the number
a
k
−
22
a_k-22
a
k
−
22
is composite.
4
1
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Functional Equation - $f(f(x)^2+f(y)) = xf(x)+y$
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
2
+
f
(
y
)
)
=
x
f
(
x
)
+
y
f(f(x)^2+f(y)) = xf(x)+y
f
(
f
(
x
)
2
+
f
(
y
))
=
x
f
(
x
)
+
y
,
∀
x
,
y
∈
R
\forall x,y\in R
∀
x
,
y
∈
R
.
3
1
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Perpendicular diagonals in convex quadrilateral
Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?
2
1
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Cyclic product of $1/a_i^2-1$
Given positive real numbers
a
1
,
a
2
,
.
.
.
,
a
n
{a_1},{a_2},...,{a_n}
a
1
,
a
2
,
...
,
a
n
with
a
1
+
a
2
+
.
.
.
+
a
n
=
1
{a_1}+{a_2}+...+{a_n}= 1
a
1
+
a
2
+
...
+
a
n
=
1
. Prove that
(
1
a
1
2
−
1
)
(
1
a
2
2
−
1
)
.
.
.
(
1
a
n
2
−
1
)
⩾
(
n
2
−
1
)
n
\left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n}
(
a
1
2
1
−
1
)
(
a
2
2
1
−
1
)
...
(
a
n
2
1
−
1
)
⩾
(
n
2
−
1
)
n
.
1
1
Hide problems
Only one solution to 1/x - 1/y = 1/n
Prove that
n
n
n
must be prime in order to have only one solution to the equation
1
x
−
1
y
=
1
n
\frac{1}{x}-\frac{1}{y}=\frac{1}{n}
x
1
−
y
1
=
n
1
,
x
,
y
∈
N
x,y\in\mathbb{N}
x
,
y
∈
N
.