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Contests
National and Regional Contests
Russia Contests
Junior Tuymaada Olympiad
2006 Junior Tuymaada Olympiad
2006 Junior Tuymaada Olympiad
Part of
Junior Tuymaada Olympiad
Subcontests
(8)
8
1
Hide problems
colouring squares (junior edition)
From a
8
×
7
8\times 7
8
×
7
rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has
14
14
14
unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with
k
k
k
colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of
k
k
k
.
7
1
Hide problems
prove PQ> AC related to intersections of lines with circumcircles
The median
B
M
BM
BM
of a triangle
A
B
C
ABC
A
BC
intersects the circumscribed circle at point
K
K
K
. The circumcircle of the triangle
K
M
C
KMC
K
MC
intersects the segment
B
C
BC
BC
at point
P
P
P
, and the circumcircle of
A
M
K
AMK
A
M
K
intersects the extension of
B
A
BA
B
A
at
Q
Q
Q
. Prove that
P
Q
>
A
C
PQ> AC
PQ
>
A
C
.
6
1
Hide problems
palindromic partitionings of 2006
Palindromic partitioning of the natural number
A
A
A
is called, when
A
A
A
is written as the sum of natural the terms
A
=
a
1
+
a
2
+
l
d
o
t
s
+
a
n
−
1
+
a
n
A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n
A
=
a
1
+
a
2
+
l
d
o
t
s
+
a
n
−
1
+
a
n
(
n
≥
1
n \geq 1
n
≥
1
), in which
a
1
=
a
n
,
a
2
=
a
n
−
1
a_1 = a_n , a_2 = a_ {n-1}
a
1
=
a
n
,
a
2
=
a
n
−
1
and in general,
a
i
=
a
n
+
1
−
i
a_i = a_ {n + 1 - i}
a
i
=
a
n
+
1
−
i
with
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
. For example,
16
=
16
16 = 16
16
=
16
,
16
=
2
+
12
+
2
16 = 2 + 12 + 2
16
=
2
+
12
+
2
and
16
=
7
+
1
+
1
+
7
16 = 7 + 1 + 1 + 7
16
=
7
+
1
+
1
+
7
are palindromic partitions of the number
16
16
16
. Find the number of all palindromic partitions of the number
2006
2006
2006
.
5
1
Hide problems
trinomials f,g,h are sides of a triangle, but f(x)-1,g(x) -1, h(x) -1 are not
The quadratic trinomials
f
f
f
,
g
g
g
and
h
h
h
are such that for every real
x
x
x
the numbers
f
(
x
)
f (x)
f
(
x
)
,
g
(
x
)
g (x)
g
(
x
)
and
h
(
x
)
h (x)
h
(
x
)
are the lengths of the sides of some triangles, and the numbers
f
(
x
)
−
1
f (x) -1
f
(
x
)
−
1
,
g
(
x
)
−
1
g (x) -1
g
(
x
)
−
1
and
h
(
x
)
−
1
h (x) -1
h
(
x
)
−
1
are not the lengths of the sides of the triangle. Prove that at least of the polynomials
f
+
g
−
h
f + g-h
f
+
g
−
h
,
f
+
h
−
g
f + h-g
f
+
h
−
g
,
g
+
h
−
f
g + h-f
g
+
h
−
f
is constant.
4
1
Hide problems
1/(x ^ 2+y+z) +1/(x+y^2+z) + 1/(x+y+z^2) <=1, if x+y+z=3, x,y,z >=0
The sum of non-negative numbers
x
x
x
,
y
y
y
and
z
z
z
is
3
3
3
. Prove the inequality
1
x
2
+
y
+
z
+
1
x
+
y
2
+
z
+
1
x
+
y
+
z
2
≤
1.
{1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1.
x
2
+
y
+
z
1
+
x
+
y
2
+
z
1
+
x
+
y
+
z
2
1
≤
1.
3
1
Hide problems
no of triangles of area 1 at vertices of convex n-gon <= 1 /3 n (2n-5)
Given a convex
n
n
n
-gon (
n
≥
5
n \geq 5
n
≥
5
). Prove that the number of triangles of area
1
1
1
with vertices at the vertices of the
n
n
n
-gon does not exceed
1
3
n
(
2
n
−
5
)
\frac{1}{3} n (2n-5)
3
1
n
(
2
n
−
5
)
.
2
1
Hide problems
difference of 16th powers of 2 odd primes to be divisible by 8 odd primes
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?
1
1
Hide problems
equal segments wanted starting with an isosceles and right triangle
On the equal
A
C
AC
A
C
and
B
C
BC
BC
of an isosceles right triangle
A
B
C
ABC
A
BC
, points
D
D
D
and
E
E
E
are marked respectively, so that
C
D
=
C
E
CD = CE
C
D
=
CE
. Perpendiculars on the straight line
A
E
AE
A
E
, passing through the points
C
C
C
and
D
D
D
, intersect the side
A
B
AB
A
B
at the points
P
P
P
and
Q
Q
Q
.Prove that
B
P
=
P
Q
BP = PQ
BP
=
PQ
.