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Problems
Contests
National and Regional Contests
Russia Contests
Junior Tuymaada Olympiad
2005 Junior Tuymaada Olympiad
2005 Junior Tuymaada Olympiad
Part of
Junior Tuymaada Olympiad
Subcontests
(7)
3
1
Hide problems
20 passengers with coins of 2 and 5
Tram ticket costs
1
1
1
Tug (
=
100
=100
=
100
tugriks).
20
20
20
passengers have only coins in denominations of
2
2
2
and
5
5
5
tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?
8
1
Hide problems
constructing sequences by adding simple divisors, common member wanted
The sequence of natural numbers is based on the following rule: each term, starting with the second, is obtained from the previous addition works of all its various simple divisors (for example, after the number
12
12
12
should be the number
18
18
18
, and after the number
125
125
125
, the number
130
130
130
). Prove that any two sequences constructed in this way have a common member.
7
1
Hide problems
AB + AC = 3BC, midpoints and incircle related, <BIC_1 +<CIB_1 = 180^o
The point
I
I
I
is the center of the inscribed circle of the triangle
A
B
C
ABC
A
BC
. The points
B
1
B_1
B
1
and
C
1
C_1
C
1
are the midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. It is known that
∠
B
I
C
1
+
∠
C
I
B
1
=
18
0
∘
\angle BIC_1 + \angle CIB_1 = 180^\circ
∠
B
I
C
1
+
∠
C
I
B
1
=
18
0
∘
. Prove the equality
A
B
+
A
C
=
3
B
C
AB + AC = 3BC
A
B
+
A
C
=
3
BC
6
1
Hide problems
comparing mobile networks' length coverage
Along the direct highway Tmutarakan - Uryupinsk at points
A
1
A_1
A
1
,
A
2
A_2
A
2
,
…
\dots
…
,
A
100
A_ {100}
A
100
are the towers of the DPS mobile operator, and in points
B
1
B_1
B
1
,
B
2
B_2
B
2
,
…
\dots
…
,
B
100
B_ {100}
B
100
are the towers of the "Horn" company. (Tower numbering may not coincide with the order of their location along the highway.) Each tower operates at a distance of
10
10
10
km in both directions along the highway. It is known that
A
i
A
k
≥
B
i
B
k
A_iA_k \geq B_iB_k
A
i
A
k
≥
B
i
B
k
for any
i
i
i
,
k
≤
100
k \leq 100
k
≤
100
. Prove that the total length of all sections of the highway covered by the DPS network is not less than the length of the sections covered by the Horn network .
5
1
Hide problems
integer f(x) = x ^ 2 + ax + b, with f (x) >= - 9/ 10, Prove f (x)>= - 1/4
Given the quadratic trinomial
f
(
x
)
=
x
2
+
a
x
+
b
f (x) = x ^ 2 + ax + b
f
(
x
)
=
x
2
+
a
x
+
b
with integer coefficients, satisfying the inequality
f
(
x
)
≥
−
9
10
f (x) \geq - {9 \over 10}
f
(
x
)
≥
−
10
9
for any
x
x
x
. Prove that
f
(
x
)
≥
−
1
4
f (x) \geq - {1 \over 4}
f
(
x
)
≥
−
4
1
for any
x
x
x
.
2
1
Hide problems
area of PKQI is half the area of ABC, incircle related
Points
X
X
X
and
Y
Y
Y
are the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
,
I
I
I
is the center of its inscribed circle,
K
K
K
is the point of tangency of the inscribed circles with side
B
C
BC
BC
. The external angle bisector at the vertex
B
B
B
intersects the line
X
Y
XY
X
Y
at the point
P
P
P
, and the external angle bisector at the vertex of
C
C
C
intersects
X
Y
XY
X
Y
at
Q
Q
Q
. Prove that the area of the quadrilateral
P
K
Q
I
PKQI
P
K
Q
I
is equal to half the area of the triangle
A
B
C
ABC
A
BC
.
1
1
Hide problems
numbers 1,2,3 in each cell of table 3x3, greatest no of different sums
In each cell of the table
3
×
3
3 \times 3
3
×
3
there is one of the numbers
1
,
2
1, 2
1
,
2
and
3
3
3
. Dima counted the sum of the numbers in each row and in each column. What is the greatest number of different sums he could get?