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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2002 National Olympiad First Round
2002 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
Hide problems
P36 [Algebra] - Turkish NMO 1st Round - 2002
If
a
5
+
5
a
4
+
10
a
3
+
3
a
2
−
9
a
−
6
=
0
a^5 +5a^4 +10a^3 +3a^2 -9a-6 = 0
a
5
+
5
a
4
+
10
a
3
+
3
a
2
−
9
a
−
6
=
0
where
a
a
a
is a real number other than
−
1
-1
−
1
, what is
(
a
+
1
)
3
(a + 1)^3
(
a
+
1
)
3
?
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27
<span class='latex-bold'>a)</span>\ 1 \qquad<span class='latex-bold'>b)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>c)</span>\ 7 \qquad<span class='latex-bold'>d)</span>\ 8 \qquad<span class='latex-bold'>e)</span>\ 27
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8
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27
35
1
Hide problems
P35 [Combinatorics] - Turkish NMO 1st Round - 2002
For each integer
i
=
0
,
1
,
2
,
…
i=0,1,2, \dots
i
=
0
,
1
,
2
,
…
, there are eight balls each weighing
2
i
2^i
2
i
grams. We may place balls as much as we desire into given
n
n
n
boxes. If the total weight of balls in each box is same, what is the largest possible value of
n
n
n
?
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15
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16
<span class='latex-bold'>a)</span>\ 8 \qquad<span class='latex-bold'>b)</span>\ 10 \qquad<span class='latex-bold'>c)</span>\ 12 \qquad<span class='latex-bold'>d)</span>\ 15 \qquad<span class='latex-bold'>e)</span>\ 16
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16
34
1
Hide problems
P34 [Number Theory] - Turkish NMO 1st Round - 2002
How many positive integers
n
n
n
are there such that
3
n
2
+
3
n
+
7
3n^2 + 3n + 7
3
n
2
+
3
n
+
7
is a perfect cube?
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7
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Infinitely many
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 3 \qquad<span class='latex-bold'>d)</span>\ 7 \qquad<span class='latex-bold'>e)</span>\ \text{Infinitely many}
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7
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Infinitely many
33
1
Hide problems
P33 [Geometry] - Turkish NMO 1st Round - 2002
Let
A
B
C
D
ABCD
A
BC
D
be a rhombus such that
m
(
A
B
C
^
)
=
4
0
∘
m(\widehat{ABC}) = 40^\circ
m
(
A
BC
)
=
4
0
∘
. Let
E
E
E
be the midpoint of
[
B
C
]
[BC]
[
BC
]
and
F
F
F
be the foot of the perpendicular from
A
A
A
to
D
E
DE
D
E
. What is
m
(
D
F
C
^
)
m(\widehat{DFC})
m
(
D
FC
)
?
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5
∘
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13
5
∘
<span class='latex-bold'>a)</span>\ 100^\circ \qquad<span class='latex-bold'>b)</span>\ 110^\circ \qquad<span class='latex-bold'>c)</span>\ 115^\circ \qquad<span class='latex-bold'>d)</span>\ 120^\circ \qquad<span class='latex-bold'>e)</span>\ 135^\circ
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5
∘
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5
∘
32
1
Hide problems
P32 [Algebra] - Turkish NMO 1st Round - 2002
Which of the following is true if
S
=
1
1
2
+
1
2
2
+
1
3
2
+
⋯
+
1
200
1
2
+
1
200
2
2
S = \dfrac 1{1^2} + \dfrac 1{2^2} + \dfrac 1{3^2} + \cdots + \dfrac 1{2001^2} + \dfrac 1{2002^2}
S
=
1
2
1
+
2
2
1
+
3
2
1
+
⋯
+
200
1
2
1
+
200
2
2
1
?
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≤
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<
7
3
<span class='latex-bold'>a)</span>\ 1\leq S < \dfrac 43 \qquad<span class='latex-bold'>b)</span>\ \dfrac 43 \leq S < 2 \qquad<span class='latex-bold'>c)</span>\ 2 \leq S < \dfrac 73
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≤
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≤
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7
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3
≤
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2
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≤
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<span class='latex-bold'>d)</span>\ \dfrac 73 \leq S < \dfrac 52 \qquad<span class='latex-bold'>e)</span>\ \dfrac 52 \leq S < 3
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5
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2
5
≤
S
<
3
31
1
Hide problems
P31 [Combinatorics] - Turkish NMO 1st Round - 2002
The numbers
1
,
2
,
…
,
N
1, 2, \dots ,N
1
,
2
,
…
,
N
are arranged in a circle where N \geq 2. If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of
N
N
N
?
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29
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None of above
<span class='latex-bold'>a)</span>\ 18 \qquad<span class='latex-bold'>b)</span>\ 19 \qquad<span class='latex-bold'>c)</span>\ 28 \qquad<span class='latex-bold'>d)</span>\ 29 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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29
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None of above
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2002
In
△
A
B
C
\triangle ABC
△
A
BC
, angle bisector ıf
C
A
B
^
\widehat{CAB}
C
A
B
meets
B
C
BC
BC
at
L
L
L
, angle bisector of
A
B
C
^
\widehat{ABC}
A
BC
meets
A
C
AC
A
C
at
N
N
N
. Lines
A
L
AL
A
L
and
B
N
BN
BN
meet at
O
O
O
. If
∣
N
L
∣
=
3
|NL| = \sqrt 3
∣
N
L
∣
=
3
, what is
∣
O
N
∣
+
∣
O
L
∣
|ON| + |OL|
∣
ON
∣
+
∣
O
L
∣
?
<
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a
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′
l
a
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−
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o
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>
a
)
<
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>
3
3
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b
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<
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2
3
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)
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>
2
<
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−
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)
<
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>
3
<
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<
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5
<span class='latex-bold'>a)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>b)</span>\ 2\sqrt 3 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 3 \qquad<span class='latex-bold'>e)</span>\ 5
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−
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3
3
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−
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b
)
<
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>
2
3
<
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c
l
a
ss
=
′
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−
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)
<
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>
2
<
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a
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=
′
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a
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−
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o
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>
d
)
<
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>
3
<
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a
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=
′
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a
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x
−
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o
l
d
′
>
e
)
<
/
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p
an
>
5
27
1
Hide problems
P27 [Combinatorics] - Turkish NMO 1st Round - 2002
The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys?
<
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18
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>
20
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−
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>
22
<
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a
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−
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o
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)
<
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>
24
<
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<
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25
<span class='latex-bold'>a)</span>\ 18 \qquad<span class='latex-bold'>b)</span>\ 20 \qquad<span class='latex-bold'>c)</span>\ 22 \qquad<span class='latex-bold'>d)</span>\ 24 \qquad<span class='latex-bold'>e)</span>\ 25
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−
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o
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)
<
/
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>
18
<
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a
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=
′
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a
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−
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o
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d
′
>
b
)
<
/
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>
20
<
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a
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=
′
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a
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−
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o
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d
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>
c
)
<
/
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>
22
<
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a
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=
′
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a
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−
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o
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>
d
)
<
/
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>
24
<
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a
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=
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a
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−
b
o
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d
′
>
e
)
<
/
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an
>
25
26
1
Hide problems
P26 [Number Theory] - Turkish NMO 1st Round - 2002
Which of the following is the set of all perfect squares that can be written as sum of three odd composite numbers?
a)
\ \{(2k + 1)^2 : k \geq 0\}
b)
\ \{(4k + 3)^2 : k \geq 1\}
c)
\ \{(2k + 1)^2 : k \geq 3\}
d)
\ \{(4k + 1)^2 : k \geq 2\}
<
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)
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a
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>
None of above
<span class='latex-bold'>e)</span>\ \text{None of above}
<
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−
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o
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>
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)
<
/
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>
None of above
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2002
Let
E
E
E
be a point on side
[
A
D
]
[AD]
[
A
D
]
of rhombus
A
B
C
D
ABCD
A
BC
D
. Lines
A
B
AB
A
B
and
C
E
CE
CE
meet at
F
F
F
, lines
B
E
BE
BE
and
D
F
DF
D
F
meet at
G
G
G
. If
m
(
D
A
B
^
)
=
6
0
∘
m(\widehat{DAB}) = 60^\circ
m
(
D
A
B
)
=
6
0
∘
, what is
m
(
D
G
B
^
)
m(\widehat{DGB})
m
(
D
GB
)
?
<
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4
5
∘
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−
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)
<
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>
5
0
∘
<
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a
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−
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>
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)
<
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>
6
0
∘
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a
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−
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d
)
<
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>
6
5
∘
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−
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)
<
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>
7
5
∘
<span class='latex-bold'>a)</span>\ 45^\circ \qquad<span class='latex-bold'>b)</span>\ 50^\circ \qquad<span class='latex-bold'>c)</span>\ 60^\circ \qquad<span class='latex-bold'>d)</span>\ 65^\circ \qquad<span class='latex-bold'>e)</span>\ 75^\circ
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4
5
∘
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=
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)
<
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>
5
0
∘
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a
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=
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−
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>
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)
<
/
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>
6
0
∘
<
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c
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a
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=
′
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a
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x
−
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o
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>
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)
<
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>
6
5
∘
<
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=
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−
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)
<
/
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>
7
5
∘
24
1
Hide problems
P24 [Algebra] - Turkish NMO 1st Round - 2002
How many positive integers
n
n
n
are there such that the equation
⌊
7
n
+
2
3
⌋
=
⌊
7
n
+
3
3
⌋
\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor
⌊
3
7
n
+
2
⌋
=
⌊
3
7
n
+
3
⌋
does not hold?
<
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)
<
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>
0
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a
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−
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)
<
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a
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>
1
<
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p
a
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c
l
a
s
s
=
′
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a
t
e
x
−
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o
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′
>
c
)
<
/
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a
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>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
a
n
>
Infinitely many
<
s
p
a
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c
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a
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s
=
′
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a
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x
−
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o
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>
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)
<
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>
None of above
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 7 \qquad<span class='latex-bold'>d)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
<
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−
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)
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>
0
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c
l
a
ss
=
′
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a
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x
−
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o
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>
b
)
<
/
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>
1
<
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p
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c
l
a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
c
)
<
/
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p
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>
7
<
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p
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c
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a
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=
′
l
a
t
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x
−
b
o
l
d
′
>
d
)
<
/
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p
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>
Infinitely many
<
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c
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a
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=
′
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a
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x
−
b
o
l
d
′
>
e
)
<
/
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an
>
None of above
23
1
Hide problems
P23 [Combinatorics] - Turkish NMO 1st Round - 2002
What is the arithmetic mean of the smallest elements of
r
r
r
-subsets of set
{
1
,
2
,
…
,
n
}
\{1, 2, \dots , n\}
{
1
,
2
,
…
,
n
}
where 1 \leq r \leq n?
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<
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>
n
+
1
r
+
1
<
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p
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a
s
s
=
′
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−
b
o
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d
′
>
b
)
<
/
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p
a
n
>
r
(
n
+
1
)
r
+
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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>
c
)
<
/
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a
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>
n
r
r
+
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
a
n
>
r
(
n
+
1
)
(
r
+
1
)
n
<
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p
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n
c
l
a
s
s
=
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a
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−
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>
e
)
<
/
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p
a
n
>
None of above
<span class='latex-bold'>a)</span>\ \dfrac{n+1}{r+1} \qquad<span class='latex-bold'>b)</span>\ \dfrac{r(n+1)}{r+1} \qquad<span class='latex-bold'>c)</span>\ \dfrac{nr}{r+1} \qquad<span class='latex-bold'>d)</span>\ \dfrac{r(n+1)}{(r+1)n} \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
<
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−
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)
<
/
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>
r
+
1
n
+
1
<
s
p
an
c
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=
′
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a
t
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x
−
b
o
l
d
′
>
b
)
<
/
s
p
an
>
r
+
1
r
(
n
+
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
c
)
<
/
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p
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>
r
+
1
n
r
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
an
>
(
r
+
1
)
n
r
(
n
+
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
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x
−
b
o
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d
′
>
e
)
<
/
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>
None of above
22
1
Hide problems
P22 [Number Theory] - Turkish NMO 1st Round - 2002
If
2
n
2^n
2
n
divides
5
256
−
1
5^{256} - 1
5
256
−
1
, what is the largest possible value of
n
n
n
?
<
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c
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a
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s
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′
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a
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−
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o
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′
>
a
)
<
/
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a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
b
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
c
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
e
)
<
/
s
p
a
n
>
None of above
<span class='latex-bold'>a)</span>\ 8 \qquad<span class='latex-bold'>b)</span>\ 10 \qquad<span class='latex-bold'>c)</span>\ 11 \qquad<span class='latex-bold'>d)</span>\ 12 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
<
s
p
an
c
l
a
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=
′
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a
t
e
x
−
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o
l
d
′
>
a
)
<
/
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>
8
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
b
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
c
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
e
)
<
/
s
p
an
>
None of above
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2002
Let
A
1
A
2
⋯
A
10
A_1A_2 \cdots A_{10}
A
1
A
2
⋯
A
10
be a regular decagon such that
[
A
1
A
4
]
=
b
[A_1A_4]=b
[
A
1
A
4
]
=
b
and the length of the circumradius is
R
R
R
. What is the length of a side of the decagon?
<
s
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−
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a
)
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b
−
R
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2
−
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2
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R
+
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2
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b
−
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R
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2
b
−
3
R
<span class='latex-bold'>a)</span>\ b-R \qquad<span class='latex-bold'>b)</span>\ b^2-R^2 \qquad<span class='latex-bold'>c)</span>\ R+\dfrac b2 \qquad<span class='latex-bold'>d)</span>\ b-2R \qquad<span class='latex-bold'>e)</span>\ 2b-3R
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2
−
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2
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R
+
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b
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b
−
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R
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)
<
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2
b
−
3
R
20
1
Hide problems
P20 [Algebra] - Turkish NMO 1st Round - 2002
Which of the following cannot be equal to
x
2
+
y
2
x^2+y^2
x
2
+
y
2
, if
x
2
+
x
y
+
y
2
=
1
x^2 + xy + y^2 = 1
x
2
+
x
y
+
y
2
=
1
where
x
,
y
x,y
x
,
y
are real numbers?
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1
2
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1
2
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−
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)
<
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>
2
<
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a
s
s
=
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x
−
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>
d
)
<
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a
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>
3
−
3
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<
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None of above
<span class='latex-bold'>a)</span>\ \dfrac{1}{\sqrt 2} \qquad<span class='latex-bold'>b)</span>\ \dfrac 12 \qquad<span class='latex-bold'>c)</span>\ \sqrt 2 \qquad<span class='latex-bold'>d)</span>\ 3-\sqrt 3 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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1
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1
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2
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)
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>
3
−
3
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o
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>
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)
<
/
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>
None of above
19
1
Hide problems
P19 [Combinatorics] - Turkish NMO 1st Round - 2002
How many positive integers
A
A
A
are there such that if we append
3
3
3
digits to the rightmost of decimal representation of
A
A
A
, we will get a number equal to
1
+
2
+
⋯
+
A
1+2+\cdots + A
1
+
2
+
⋯
+
A
?
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0
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<
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1
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2
<
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a
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′
l
a
t
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x
−
b
o
l
d
′
>
d
)
<
/
s
p
a
n
>
2002
<
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a
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)
<
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>
None of above
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 2002 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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0
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)
<
/
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>
1
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)
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>
2
<
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p
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c
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a
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=
′
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a
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x
−
b
o
l
d
′
>
d
)
<
/
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p
an
>
2002
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−
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o
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>
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)
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None of above
18
1
Hide problems
P18 [Number Theory] - Turkish NMO 1st Round - 2002
For how many integers
x
x
x
is
∣
15
x
2
−
32
x
−
28
∣
|15x^2-32x-28|
∣15
x
2
−
32
x
−
28∣
a prime number?
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0
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−
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>
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)
<
/
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>
1
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a
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=
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−
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2
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=
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−
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d
)
<
/
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>
4
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None of above
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 4 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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0
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a
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−
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>
b
)
<
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1
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=
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−
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>
c
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2
<
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a
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=
′
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a
t
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x
−
b
o
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′
>
d
)
<
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>
4
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>
e
)
<
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>
None of above
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2002
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid and a tangential quadrilateral such that
A
D
∣
∣
B
C
AD || BC
A
D
∣∣
BC
and
∣
A
B
∣
=
∣
C
D
∣
|AB|=|CD|
∣
A
B
∣
=
∣
C
D
∣
. The incircle touches
[
C
D
]
[CD]
[
C
D
]
at
N
N
N
.
[
A
N
]
[AN]
[
A
N
]
and
[
B
N
]
[BN]
[
BN
]
meet the incircle again at
K
K
K
and
L
L
L
, respectively. What is
∣
A
N
∣
∣
A
K
∣
+
∣
B
N
∣
∣
B
L
∣
\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}
∣
A
K
∣
∣
A
N
∣
+
∣
B
L
∣
∣
BN
∣
?
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(
A
)
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8
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(
B
)
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>
9
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(
C
)
<
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10
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(
D
)
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12
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(
E
)
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16
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 9 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 16
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A
)
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8
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(
B
)
<
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9
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(
C
)
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10
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(
D
)
<
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>
12
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(
E
)
<
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>
16
16
1
Hide problems
P16 [Algebra] - Turkish NMO 1st Round - 2002
Which of the following cannot be equal to
x
2
+
1
4
x
x^2 + \dfrac 1{4x}
x
2
+
4
x
1
where
x
x
x
is a positive real number?
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3
−
1
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−
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o
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b
)
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>
2
2
−
2
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−
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c
)
<
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>
5
−
1
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−
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)
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1
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)
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None of above
<span class='latex-bold'>a)</span>\ \sqrt 3 -1 \qquad<span class='latex-bold'>b)</span>\ 2\sqrt 2 - 2 \qquad<span class='latex-bold'>c)</span>\ \sqrt 5 - 1 \qquad<span class='latex-bold'>d)</span>\ 1 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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3
−
1
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−
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o
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)
<
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>
2
2
−
2
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−
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c
)
<
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5
−
1
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1
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−
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o
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>
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)
<
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an
>
None of above
15
1
Hide problems
P15 [Combinatorics] - Turkish NMO 1st Round - 2002
There are
10
10
10
seats in each of
10
10
10
rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats?
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1
55
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)
<
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>
1
50
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<
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2
55
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−
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o
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d
)
<
/
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a
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>
1
25
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=
′
l
a
t
e
x
−
b
o
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d
′
>
e
)
<
/
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a
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>
None of above
<span class='latex-bold'>a)</span>\ \dfrac{1}{55} \qquad<span class='latex-bold'>b)</span>\ \dfrac{1}{50} \qquad<span class='latex-bold'>c)</span>\ \dfrac{2}{55} \qquad<span class='latex-bold'>d)</span>\ \dfrac{1}{25} \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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55
1
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50
1
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2
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25
1
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None of above
14
1
Hide problems
P14 [Number Theory] - Turkish NMO 1st Round - 2002
How many primes
p
p
p
are there such that
39
p
+
1
39p + 1
39
p
+
1
is a perfect square?
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0
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1
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2
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3
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None of above
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 3 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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None of above
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2002
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
∣
B
C
∣
+
∣
A
D
∣
=
7
|BC|+|AD| = 7
∣
BC
∣
+
∣
A
D
∣
=
7
,
∣
A
B
∣
=
9
|AB| = 9
∣
A
B
∣
=
9
and
∣
B
C
∣
=
14
|BC| = 14
∣
BC
∣
=
14
. What is the ratio of the area of the triangle formed by
C
D
CD
C
D
, angle bisector of
B
C
D
^
\widehat{BCD}
BC
D
and angle bisector of
C
D
A
^
\widehat{CDA}
C
D
A
over the area of the trapezoid?
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9
14
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7
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49
69
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1
3
<span class='latex-bold'>a)</span>\ \dfrac{9}{14} \qquad<span class='latex-bold'>b)</span>\ \dfrac{5}{7} \qquad<span class='latex-bold'>c)</span>\ \sqrt 2 \qquad<span class='latex-bold'>d)</span>\ \dfrac{49}{69} \qquad<span class='latex-bold'>e)</span>\ \dfrac 13
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9
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5
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69
49
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3
1
12
1
Hide problems
P12 [Algebra] - Turkish NMO 1st Round - 2002
What is the least possible value of
a
b
+
b
c
+
a
c
ab + bc + ac
ab
+
b
c
+
a
c
such that
a
2
+
b
2
+
c
2
=
1
a^2 + b^2 + c^2 = 1
a
2
+
b
2
+
c
2
=
1
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers?
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1
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2
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3
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2
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0
<span class='latex-bold'>a)</span>\ -1 \qquad<span class='latex-bold'>b)</span>\ -\dfrac 12 \qquad<span class='latex-bold'>c)</span>\ -\dfrac 13 \qquad<span class='latex-bold'>d)</span>\ -\dfrac{1}{2\sqrt 2} \qquad<span class='latex-bold'>e)</span>\ 0
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1
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0
11
1
Hide problems
P11 [Combinatorics] - Turkish NMO 1st Round - 2002
What is the coefficient of
x
5
x^5
x
5
in the expansion of
(
1
+
x
+
x
2
)
9
(1 + x + x^2)^9
(
1
+
x
+
x
2
)
9
?
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>
1680
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882
<
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−
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<
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>
729
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450
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246
<span class='latex-bold'>a)</span>\ 1680 \qquad<span class='latex-bold'>b)</span>\ 882 \qquad<span class='latex-bold'>c)</span>\ 729 \qquad<span class='latex-bold'>d)</span>\ 450 \qquad<span class='latex-bold'>e)</span>\ 246
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1680
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)
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882
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)
<
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729
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450
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246
10
1
Hide problems
P10 [Number Theory] - Turkish NMO 1st Round - 2002
Which of the following does not divide the number of ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of integers satisfying the equation
x
3
−
13
y
3
=
1453
x^3 - 13y^3 = 1453
x
3
−
13
y
3
=
1453
?
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2
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3
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5
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7
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None of above
<span class='latex-bold'>a)</span>\ 2 \qquad<span class='latex-bold'>b)</span>\ 3 \qquad<span class='latex-bold'>c)</span>\ 5 \qquad<span class='latex-bold'>d)</span>\ 7 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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5
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7
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None of above
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2002
Let
A
B
C
ABC
A
BC
be triangle such that
∣
A
B
∣
=
5
|AB| = 5
∣
A
B
∣
=
5
,
∣
B
C
∣
=
9
|BC| = 9
∣
BC
∣
=
9
and
∣
A
C
∣
=
8
|AC| = 8
∣
A
C
∣
=
8
. The angle bisector of
B
C
A
^
\widehat{BCA}
BC
A
meets
B
A
BA
B
A
at
X
X
X
and the angle bisector of
C
A
B
^
\widehat{CAB}
C
A
B
meets
B
C
BC
BC
at
Y
Y
Y
. Let
Z
Z
Z
be the intersection of lines
X
Y
XY
X
Y
and
A
C
AC
A
C
. What is
∣
A
Z
∣
|AZ|
∣
A
Z
∣
?
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104
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145
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89
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9
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10
<span class='latex-bold'>a)</span>\ \sqrt{104} \qquad<span class='latex-bold'>b)</span>\ \sqrt{145} \qquad<span class='latex-bold'>c)</span>\ \sqrt{89} \qquad<span class='latex-bold'>d)</span>\ 9 \qquad<span class='latex-bold'>e)</span>\ 10
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104
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a
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−
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)
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145
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89
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9
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10
8
1
Hide problems
P08 [Algebra] - Turkish NMO 1st Round - 2002
Which of the following polynomials does not divide
x
60
−
1
x^{60} - 1
x
60
−
1
?
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x
2
+
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+
1
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4
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1
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5
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15
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None of above
<span class='latex-bold'>a)</span>\ x^2+x+1 \qquad<span class='latex-bold'>b)</span>\ x^4-1 \qquad<span class='latex-bold'>c)</span>\ x^5-1 \qquad<span class='latex-bold'>d)</span>\ x^{15}-1 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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2
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4
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15
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None of above
7
1
Hide problems
P07 [Combinatorics] - Turkish NMO 1st Round - 2002
What is the least number of weighings needed to determine the sum of weights of
13
13
13
watermelons such that exactly two watermelons should be weighed in each weigh?
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11
<span class='latex-bold'>a)</span>\ 7 \qquad<span class='latex-bold'>b)</span>\ 8 \qquad<span class='latex-bold'>c)</span>\ 9 \qquad<span class='latex-bold'>d)</span>\ 10 \qquad<span class='latex-bold'>e)</span>\ 11
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7
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8
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11
6
1
Hide problems
P06 [Number Theory] - Turkish NMO 1st Round - 2002
The thousands digit of a five-digit number which is divisible by
37
37
37
and
173
173
173
is
3
3
3
. What is the hundreds digit of this number?
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8
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 2 \qquad<span class='latex-bold'>c)</span>\ 4 \qquad<span class='latex-bold'>d)</span>\ 6 \qquad<span class='latex-bold'>e)</span>\ 8
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8
5
1
Hide problems
P05 [Geometry] - Turkish NMO 1st Round - 2002
The lengths of two altitudes of a triangles are
8
8
8
and
12
12
12
. Which of the following cannot be the third altitude?
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7
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12
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<span class='latex-bold'>a)</span>\ 4 \qquad<span class='latex-bold'>b)</span>\ 7 \qquad<span class='latex-bold'>c)</span>\ 8 \qquad<span class='latex-bold'>d)</span>\ 12 \qquad<span class='latex-bold'>e)</span>\ 23
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7
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23
4
1
Hide problems
P04 [Algebra] - Turkish NMO 1st Round - 2002
How many real roots does the polynomial
x
5
+
x
4
−
x
3
−
x
2
−
2
x
−
2
x^5 + x^4 - x^3 - x^2 - 2x - 2
x
5
+
x
4
−
x
3
−
x
2
−
2
x
−
2
have?
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1
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4
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None of above
<span class='latex-bold'>a)</span>\ 1 \qquad<span class='latex-bold'>b)</span>\ 2 \qquad<span class='latex-bold'>c)</span>\ 3 \qquad<span class='latex-bold'>d)</span>\ 4 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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4
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None of above
3
1
Hide problems
P03 [Combinatorics] - Turkish NMO 1st Round - 2002
In the beginnig, each unit square of a
m
×
n
m\times n
m
×
n
board is colored white. We are supposed to color all the squares such that one of two adjacent squares having a common side is black and the other is white. At each move, we choose a
2
×
2
2 \times 2
2
×
2
square, and we color each of
4
4
4
squares inversely such that if the square is black then it is colored white or vice versa. For which of the following ordered pair
(
m
,
n
)
(m, n)
(
m
,
n
)
, can the board be colored in this way?
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(
3
,
3
)
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(
2
,
6
)
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(
4
,
8
)
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(
5
,
5
)
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None of above
<span class='latex-bold'>a)</span>\ (3,3) \qquad<span class='latex-bold'>b)</span>\ (2,6) \qquad<span class='latex-bold'>c)</span>\ (4,8) \qquad<span class='latex-bold'>d)</span>\ (5,5) \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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(
3
,
3
)
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(
2
,
6
)
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(
4
,
8
)
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(
5
,
5
)
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None of above
2
1
Hide problems
P02 [Number Theory] - Turkish NMO 1st Round - 2002
What is
3
2002
3^{2002}
3
2002
in
m
o
d
11
\bmod 11
mod
11
?
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3
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5
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None of above
<span class='latex-bold'>a)</span>\ 1 \qquad<span class='latex-bold'>b)</span>\ 3 \qquad<span class='latex-bold'>c)</span>\ 4 \qquad<span class='latex-bold'>d)</span>\ 5 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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None of above
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2002
Let
C
′
,
A
′
,
B
′
C', A', B'
C
′
,
A
′
,
B
′
be the midpoints of sides
[
A
B
]
[AB]
[
A
B
]
,
[
B
C
]
[BC]
[
BC
]
,
[
C
A
]
[CA]
[
C
A
]
of
△
A
B
C
\triangle ABC
△
A
BC
, respectively. Let
H
H
H
be the foot of perpendicular from
A
A
A
to
B
C
BC
BC
. If
∣
A
′
C
′
∣
=
6
|A'C'| = 6
∣
A
′
C
′
∣
=
6
, what is
∣
B
′
H
∣
|B'H|
∣
B
′
H
∣
?
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7
<span class='latex-bold'>a)</span>\ 5 \qquad<span class='latex-bold'>b)</span>\ 6 \qquad<span class='latex-bold'>c)</span>\ 5\sqrt 2 \qquad<span class='latex-bold'>d)</span>\ 6\sqrt 2 \qquad<span class='latex-bold'>e)</span>\ 7
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7
28
1
Hide problems
Number of positive roots of polynomial with integer coeff
How many positive roots does polynomial
x
2002
+
a
2001
x
2001
+
a
2000
x
2000
+
⋯
+
a
1
x
+
a
0
x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0
x
2002
+
a
2001
x
2001
+
a
2000
x
2000
+
⋯
+
a
1
x
+
a
0
have such that
a
2001
=
2002
a_{2001} = 2002
a
2001
=
2002
and
a
k
=
−
k
−
1
a_k = -k - 1
a
k
=
−
k
−
1
for 0\leq k \leq 2000?
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1001
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2002
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 1001 \qquad<span class='latex-bold'>e)</span>\ 2002
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1001
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2002
30
1
Hide problems
P30 [Number Theory] - Turkish NMO 1st Round - 2002
How many integers 0 \leq x < 125 are there such that x^3 - 2x + 6 \equiv 0 \pmod {125}?
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None of above
<span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 3 \qquad<span class='latex-bold'>e)</span>\ \text{None of above}
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3
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None of above