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National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
2014 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(32)
32
1
Hide problems
P32 [Combinatorics] - Turkish NMO 1st Round - 2014
There are
k
k
k
stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each
k
=
5
,
6
,
7
,
8
,
9
k=5,6,7,8,9
k
=
5
,
6
,
7
,
8
,
9
. If Alper is always the first player, for how many of the games can Alper guarantee that he does not lose the game?
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5
<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>\ 5
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5
31
1
Hide problems
P31 [Algebra] - Turkish NMO 1st Round - 2014
a
1
=
1
a_{1}=1
a
1
=
1
and for all
n
≥
1
n \geq 1
n
≥
1
,
(
a
n
+
1
−
2
a
n
)
⋅
(
a
n
+
1
−
1
a
n
+
2
)
=
0.
(a_{n+1}-2a_{n})\cdot \left (a_{n+1} - \dfrac{1}{a_{n}+2} \right )=0.
(
a
n
+
1
−
2
a
n
)
⋅
(
a
n
+
1
−
a
n
+
2
1
)
=
0.
If
a
k
=
1
a_{k}=1
a
k
=
1
, which of the following can be equal to
k
k
k
?
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6
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(
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8
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>
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C
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10
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s
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(
D
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12
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None of the preceding
<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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A
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(
B
)
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8
<
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>
(
C
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10
<
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a
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=
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>
(
D
)
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12
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>
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E
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>
None of the preceding
30
1
Hide problems
P30 [Number Theory] - Turkish NMO 1st Round - 2014
Let
s
(
n
)
s(n)
s
(
n
)
denote the number of positive divisors of positive integer
n
n
n
. What is the largest prime divisor of the sum of numbers
(
s
(
k
)
)
3
(s(k))^3
(
s
(
k
)
)
3
for all positive divisors
k
k
k
of
201
4
2014
2014^{2014}
201
4
2014
?
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7
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11
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s
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D
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13
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None of the preceding
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 11 \qquad<span class='latex-bold'>(D)</span>\ 13 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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A
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5
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−
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>
(
B
)
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7
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(
D
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13
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(
E
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<
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>
None of the preceding
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2014
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
B
∣
=
13
,
∣
B
C
∣
=
12
|AB|=13 , |BC|=12
∣
A
B
∣
=
13
,
∣
BC
∣
=
12
and
∣
C
A
∣
=
5
|CA|=5
∣
C
A
∣
=
5
. Let the angle bisectors of
A
A
A
and
B
B
B
intersect at
I
I
I
and meet the opposing sides at
D
D
D
and
E
E
E
, respectively. The line passing through
I
I
I
and the midpoint of
[
D
E
]
[DE]
[
D
E
]
meets
[
A
B
]
[AB]
[
A
B
]
at
F
F
F
. What is
∣
A
F
∣
|AF|
∣
A
F
∣
?
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2
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>
(
C
)
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2
<
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s
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−
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(
D
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3
<
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(
E
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7
2
<span class='latex-bold'>(A)</span>\ \dfrac{3}{2} \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ \dfrac{5}{2} \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \dfrac{7}{2}
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2
3
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−
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>
(
B
)
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2
<
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(
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5
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(
D
)
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3
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=
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−
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>
(
E
)
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2
7
26
1
Hide problems
P26 [Number Theory] - Turkish NMO 1st Round - 2014
Let
f
(
n
)
f(n)
f
(
n
)
be the smallest prime which divides
n
4
+
1
n^4+1
n
4
+
1
. What is the remainder when the sum
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
2014
)
f(1)+f(2)+\cdots+f(2014)
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
2014
)
is divided by
8
8
8
?
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(
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<
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>
(
D
)
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7
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−
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E
)
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None of the preceding
<span class='latex-bold'>(A)</span>\ 1 \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 the preceding}
<
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(
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)
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>
1
<
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−
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>
(
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)
<
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>
3
<
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(
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)
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<
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p
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a
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=
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t
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x
−
b
o
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d
′
>
(
D
)
<
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>
7
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=
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x
−
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o
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d
′
>
(
E
)
<
/
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an
>
None of the preceding
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2014
The circle
C
1
C_{1}
C
1
with radius
6
6
6
and the circle
C
2
C_{2}
C
2
with radius
8
8
8
are externally tangent to each other at
A
A
A
. The circle
C
3
C_3
C
3
which is externally tangent to
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
has a radius with length
21
21
21
. The common tangent of
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
which passes through
A
A
A
meets
C
3
C_{3}
C
3
at
B
B
B
and
C
C
C
. What is
∣
B
C
∣
|BC|
∣
BC
∣
?
<
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(
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)
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24
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(
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)
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25
<
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=
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−
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>
(
C
)
<
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14
3
<
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c
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a
s
s
=
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t
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x
−
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o
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′
>
(
D
)
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>
24
3
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25
3
<span class='latex-bold'>(A)</span>\ 24 \qquad<span class='latex-bold'>(B)</span>\ 25 \qquad<span class='latex-bold'>(C)</span>\ 14\sqrt{3} \qquad<span class='latex-bold'>(D)</span>\ 24\sqrt{3} \qquad<span class='latex-bold'>(E)</span>\ 25\sqrt{3}
<
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−
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>
(
B
)
<
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>
25
<
s
p
an
c
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a
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=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
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>
14
3
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(
D
)
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24
3
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(
E
)
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25
3
23
1
Hide problems
P23 [Algebra] - Turkish NMO 1st Round - 2014
What is the minimum value of
(
x
2
+
2
x
+
8
−
4
3
)
⋅
(
x
2
−
6
x
+
16
−
4
3
)
(x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})
(
x
2
+
2
x
+
8
−
4
3
)
⋅
(
x
2
−
6
x
+
16
−
4
3
)
where
x
x
x
is a real number?
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(
A
)
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112
−
64
3
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l
a
s
s
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−
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o
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>
(
B
)
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3
−
3
<
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l
a
s
s
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t
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x
−
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o
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(
C
)
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8
−
4
3
<
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n
c
l
a
s
s
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x
−
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(
D
)
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>
3
3
−
4
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(
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None of the preceding
<span class='latex-bold'>(A)</span>\ 112-64\sqrt{3} \qquad<span class='latex-bold'>(B)</span>\ 3-\sqrt{3} \qquad<span class='latex-bold'>(C)</span>\ 8-4\sqrt{3} \\ <span class='latex-bold'>(D)</span>\ 3\sqrt{3}-4 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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A
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112
−
64
3
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−
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(
B
)
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3
−
3
<
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(
C
)
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8
−
4
3
<
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ss
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x
−
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(
D
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3
3
−
4
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(
E
)
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>
None of the preceding
22
1
Hide problems
P22 [Number Theory] - Turkish NMO 1st Round - 2014
What is remainder when
201
4
2015
2014^{2015}
201
4
2015
is divided by
121
121
121
?
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(
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)
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45
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(
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)
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>
34
<
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23
<
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(
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12
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1
<span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 34 \qquad<span class='latex-bold'>(C)</span>\ 23 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 1
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45
<
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(
B
)
<
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34
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(
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23
<
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(
D
)
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12
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>
(
E
)
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1
24
1
Hide problems
1,2,..n divided into 2 groups
If the integers
1
,
2
,
…
,
n
1,2,\dots,n
1
,
2
,
…
,
n
can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of
n
n
n
?
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(
A
)
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7
<
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>
(
B
)
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>
8
<
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a
s
s
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>
9
<
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a
s
s
=
′
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t
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−
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(
D
)
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>
10
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(
E
)
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None of the preceding
<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>\ \text{None of the preceding}
<
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(
A
)
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>
7
<
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p
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c
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a
ss
=
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a
t
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x
−
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>
(
B
)
<
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>
8
<
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a
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a
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−
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′
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(
C
)
<
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>
9
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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>
10
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=
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>
(
E
)
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>
None of the preceding
27
1
Hide problems
P27 [Algebra] - Turkish NMO 1st Round - 2014
Let
f
f
f
be a function defined on positive integers such that
f
(
1
)
=
4
f(1)=4
f
(
1
)
=
4
,
f
(
2
n
)
=
f
(
n
)
f(2n)=f(n)
f
(
2
n
)
=
f
(
n
)
and
f
(
2
n
+
1
)
=
f
(
n
)
+
2
f(2n+1)=f(n)+2
f
(
2
n
+
1
)
=
f
(
n
)
+
2
for every positive integer
n
n
n
. For how many positive integers
k
k
k
less than
2014
2014
2014
, it is
f
(
k
)
=
8
f(k)=8
f
(
k
)
=
8
?
<
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(
A
)
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45
<
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c
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s
=
′
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a
t
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x
−
b
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′
>
(
B
)
<
/
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a
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>
120
<
s
p
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
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>
(
C
)
<
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s
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a
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>
165
<
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l
a
s
s
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x
−
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o
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(
D
)
<
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>
180
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)
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215
<span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 120 \qquad<span class='latex-bold'>(C)</span>\ 165 \qquad<span class='latex-bold'>(D)</span>\ 180 \qquad<span class='latex-bold'>(E)</span>\ 215
<
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)
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45
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c
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a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
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>
120
<
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c
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a
ss
=
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a
t
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x
−
b
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>
(
C
)
<
/
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>
165
<
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l
a
ss
=
′
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a
t
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x
−
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d
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>
(
D
)
<
/
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>
180
<
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p
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c
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a
ss
=
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x
−
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d
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>
(
E
)
<
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>
215
28
1
Hide problems
Delete a,b; Write 2a+b, 2b+a
The integers
−
1
-1
−
1
,
2
2
2
,
−
3
-3
−
3
,
4
4
4
,
−
5
-5
−
5
,
6
6
6
are written on a blackboard. At each move, we erase two numbers
a
a
a
and
b
b
b
, then we re-write
2
a
+
b
2a+b
2
a
+
b
and
2
b
+
a
2b+a
2
b
+
a
. How many of the sextuples
(
0
,
0
,
0
,
3
,
−
9
,
9
)
(0,0,0,3,-9,9)
(
0
,
0
,
0
,
3
,
−
9
,
9
)
,
(
0
,
1
,
1
,
3
,
6
,
−
6
)
(0,1,1,3,6,-6)
(
0
,
1
,
1
,
3
,
6
,
−
6
)
,
(
0
,
0
,
0
,
3
,
−
6
,
9
)
(0,0,0,3,-6,9)
(
0
,
0
,
0
,
3
,
−
6
,
9
)
,
(
0
,
1
,
1
,
−
3
,
6
,
−
9
)
(0,1,1,-3,6,-9)
(
0
,
1
,
1
,
−
3
,
6
,
−
9
)
,
(
0
,
0
,
2
,
5
,
5
,
6
)
(0,0,2,5,5,6)
(
0
,
0
,
2
,
5
,
5
,
6
)
can be gotten?
<
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)
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1
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s
=
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−
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>
(
B
)
<
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a
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>
2
<
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c
l
a
s
s
=
′
l
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t
e
x
−
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>
(
C
)
<
/
s
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a
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>
3
<
s
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
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>
(
D
)
<
/
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a
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>
4
<
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s
s
=
′
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a
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>
(
E
)
<
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>
5
<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>\ 5
<
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−
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(
A
)
<
/
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>
1
<
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p
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c
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a
ss
=
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a
t
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−
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d
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>
(
B
)
<
/
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>
2
<
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ss
=
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−
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d
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>
(
C
)
<
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>
3
<
s
p
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c
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a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
D
)
<
/
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>
4
<
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c
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a
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=
′
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a
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−
b
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d
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>
(
E
)
<
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>
5
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2014
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that side
[
A
B
]
[AB]
[
A
B
]
and side
[
C
D
]
[CD]
[
C
D
]
are perpendicular to side
[
B
C
]
[BC]
[
BC
]
. Let
E
E
E
be a point on side
[
B
C
]
[BC]
[
BC
]
such that
△
A
E
D
\triangle AED
△
A
E
D
is equilateral. If
∣
A
B
∣
=
7
|AB|=7
∣
A
B
∣
=
7
and
∣
C
D
∣
=
5
|CD|=5
∣
C
D
∣
=
5
, what is the area of trapezoid
A
B
C
D
ABCD
A
BC
D
?
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)
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27
3
<
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(
B
)
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42
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24
3
<
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c
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a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
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40
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<span class='latex-bold'>(A)</span>\ 27\sqrt{3} \qquad<span class='latex-bold'>(B)</span>\ 42 \qquad<span class='latex-bold'>(C)</span>\ 24\sqrt{3} \qquad<span class='latex-bold'>(D)</span>\ 40 \qquad<span class='latex-bold'>(E)</span>\ 36
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3
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42
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3
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40
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36
20
1
Hide problems
Set contains powers of 2 or 3
How many distinct sets are there such that each set contains only non-negative powers of
2
2
2
or
3
3
3
and sum of its elements is
2014
2014
2014
?
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64
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60
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54
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48
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None of the preceding
<span class='latex-bold'>(A)</span>\ 64 \qquad<span class='latex-bold'>(B)</span>\ 60 \qquad<span class='latex-bold'>(C)</span>\ 54 \qquad<span class='latex-bold'>(D)</span>\ 48 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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64
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60
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54
<
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(
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48
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None of the preceding
19
1
Hide problems
P19 [Algebra] - Turkish NMO 1st Round - 2014
What is the largest possible value of
x
2
+
2
x
+
6
x
2
+
x
+
5
\dfrac{x^2+2x+6}{x^2+x+5}
x
2
+
x
+
5
x
2
+
2
x
+
6
where
x
x
x
is a positive real number?
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14
11
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9
7
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13
10
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(
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4
3
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>
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)
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None of the preceding
<span class='latex-bold'>(A)</span>\ \dfrac{14}{11} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{9}{7} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{13}{10} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{4}{3} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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11
14
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9
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13
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4
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None of the preceding
18
1
Hide problems
P18 [Number Theory] - Turkish NMO 1st Round - 2014
Which one below cannot be expressed in the form
x
2
+
y
5
x^2+y^5
x
2
+
y
5
, where
x
x
x
and
y
y
y
are integers?
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59170
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59149
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59130
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59121
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59012
<span class='latex-bold'>(A)</span>\ 59170 \qquad<span class='latex-bold'>(B)</span>\ 59149 \qquad<span class='latex-bold'>(C)</span>\ 59130 \qquad<span class='latex-bold'>(D)</span>\ 59121 \qquad<span class='latex-bold'>(E)</span>\ 59012
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59170
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59149
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59130
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59121
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59012
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2014
Let
E
E
E
be the midpoint of side
[
A
B
]
[AB]
[
A
B
]
of square
A
B
C
D
ABCD
A
BC
D
. Let the circle through
B
B
B
with center
A
A
A
and segment
[
E
C
]
[EC]
[
EC
]
meet at
F
F
F
. What is
∣
E
F
∣
/
∣
F
C
∣
|EF|/|FC|
∣
EF
∣/∣
FC
∣
?
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2
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5
−
1
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3
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3
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{2} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{5}-1 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \sqrt{3}
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3
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3
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3
16
1
Hide problems
P16 [Combinatorics] - Turkish NMO 1st Round - 2014
Asli will distribute
100
100
100
candies among her brother and
18
18
18
friends of him. Asli splits friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes same number of candies and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most how many candies can Asli's brother have?
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12
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(
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14
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16
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17
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18
<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 16 \qquad<span class='latex-bold'>(D)</span>\ 17 \qquad<span class='latex-bold'>(E)</span>\ 18
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12
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17
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)
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18
15
1
Hide problems
P15 [Algebra] - Turkish NMO 1st Round - 2014
What is the sum of distinct real numbers
x
x
x
such that
(
2
x
2
+
5
x
+
9
)
2
=
56
(
x
3
+
1
)
(2x^2+5x+9)^2=56(x^3+1)
(
2
x
2
+
5
x
+
9
)
2
=
56
(
x
3
+
1
)
?
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3
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(
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)
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7
4
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9
2
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None of the preceding
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ \dfrac{7}{4} \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{9}{2} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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3
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>
(
B
)
<
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4
7
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(
C
)
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4
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b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
14
1
Hide problems
P14 [Number Theory] - Turkish NMO 1st Round - 2014
For how many different primes
p
p
p
, there exists an integer
n
n
n
such that
p
∣
n
3
+
3
p\mid n^3+3
p
∣
n
3
+
3
and
p
∣
n
5
+
5
p\mid n^5+5
p
∣
n
5
+
5
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
Infinitely many
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
Infinitely many
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2014
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
m
(
A
D
B
^
)
=
1
5
∘
m \left (\widehat{ADB} \right)=15^{\circ}
m
(
A
D
B
)
=
1
5
∘
,
m
(
B
C
D
^
)
=
9
0
∘
m \left (\widehat{BCD} \right)=90^{\circ}
m
(
BC
D
)
=
9
0
∘
. The diagonals of quadrilateral are perpendicular at
E
E
E
. Let
P
P
P
be a point on
∣
A
E
∣
|AE|
∣
A
E
∣
such that
∣
E
C
∣
=
4
,
∣
E
A
∣
=
8
|EC|=4, |EA|=8
∣
EC
∣
=
4
,
∣
E
A
∣
=
8
and
∣
E
P
∣
=
2
|EP|=2
∣
EP
∣
=
2
. What is
m
(
P
B
D
^
)
m \left (\widehat{PBD} \right)
m
(
PB
D
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
7
5
∘
<span class='latex-bold'>(A)</span>\ 15^{\circ} \qquad<span class='latex-bold'>(B)</span>\ 30^{\circ} \qquad<span class='latex-bold'>(C)</span>\ 45^{\circ} \qquad<span class='latex-bold'>(D)</span>\ 60^{\circ} \qquad<span class='latex-bold'>(E)</span>\ 75^{\circ}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
7
5
∘
12
1
Hide problems
P12 [Combinatorics] - Turkish NMO 1st Round - 2014
If one can find a student with at least
k
k
k
friends in any class which has
21
21
21
students such that at least two of any three of these students are friends, what is the largest possible value of
k
k
k
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
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
>
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
12
<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>\ 11 \qquad<span class='latex-bold'>(E)</span>\ 12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
12
11
1
Hide problems
P11 [Algebra] - Turkish NMO 1st Round - 2014
What is the product of real numbers
a
a
a
which make
x
2
+
a
x
+
1
x^2+ax+1
x
2
+
a
x
+
1
a negative integer for only one real number
x
x
x
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
−
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
−
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
−
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
−
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
−
8
<span class='latex-bold'>(A)</span>\ -1 \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
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
−
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
−
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
−
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
−
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
−
8
10
1
Hide problems
P10 [Number Theory] - Turkish NMO 1st Round - 2014
How many non-negative integer triples
(
m
,
n
,
k
)
(m,n,k)
(
m
,
n
,
k
)
are there such that
m
3
−
n
3
=
9
k
+
123
m^3-n^3=9^k+123
m
3
−
n
3
=
9
k
+
123
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
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 the preceding
<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 the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2014
Let
D
D
D
be a point on side
[
B
C
]
[BC]
[
BC
]
of
△
A
B
C
\triangle ABC
△
A
BC
such that
∣
A
B
∣
=
3
,
∣
C
D
∣
=
1
|AB|=3, |CD|=1
∣
A
B
∣
=
3
,
∣
C
D
∣
=
1
and
∣
A
C
∣
=
∣
B
D
∣
=
5
|AC|=|BD|=\sqrt{5}
∣
A
C
∣
=
∣
B
D
∣
=
5
. If the
B
B
B
-altitude of
△
A
B
C
\triangle ABC
△
A
BC
meets
A
D
AD
A
D
at
E
E
E
, what is
∣
C
E
∣
|CE|
∣
CE
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
5
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
3
2
<span class='latex-bold'>(A)</span>\ \dfrac{2}{\sqrt{5}} \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \dfrac{2}{\sqrt{3}} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{\sqrt{5}}{2} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{3}{2}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
2
<
s
p
an
c
l
a
ss
=
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(
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2
5
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2
3
8
1
Hide problems
P08 [Combinatorics] - Turkish NMO 1st Round - 2014
In how many ways can
17
17
17
identical red and
10
10
10
identical white balls be distributed into
4
4
4
distinct boxes such that the number of red balls is greater than the number of white balls in each box?
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5462
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5586
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5664
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5720
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5848
<span class='latex-bold'>(A)</span>\ 5462 \qquad<span class='latex-bold'>(B)</span>\ 5586 \qquad<span class='latex-bold'>(C)</span>\ 5664 \qquad<span class='latex-bold'>(D)</span>\ 5720 \qquad<span class='latex-bold'>(E)</span>\ 5848
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5586
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5720
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5848
7
1
Hide problems
P07 [Algebra] - Turkish NMO 1st Round - 2014
If
(
x
2
+
1
)
(
y
2
+
1
)
+
9
=
6
(
x
+
y
)
(x^2+1)(y^2+1)+9=6(x+y)
(
x
2
+
1
)
(
y
2
+
1
)
+
9
=
6
(
x
+
y
)
where
x
,
y
x,y
x
,
y
are real numbers, what is
x
2
+
y
2
x^2+y^2
x
2
+
y
2
?
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7
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6
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3
<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 3
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3
6
1
Hide problems
P06 [Number Theory] - Turkish NMO 1st Round - 2014
The numbers which contain only even digits in their decimal representations are written in ascending order such that
2
,
4
,
6
,
8
,
20
,
22
,
24
,
26
,
28
,
40
,
42
,
…
2,4,6,8,20,22,24,26,28,40,42,\dots
2
,
4
,
6
,
8
,
20
,
22
,
24
,
26
,
28
,
40
,
42
,
…
What is the
201
4
th
2014^{\text{th}}
201
4
th
number in that sequence?
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66480
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64096
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62048
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60288
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None of the preceding
<span class='latex-bold'>(A)</span>\ 66480 \qquad<span class='latex-bold'>(B)</span>\ 64096 \qquad<span class='latex-bold'>(C)</span>\ 62048 \qquad<span class='latex-bold'>(D)</span>\ 60288 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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64096
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60288
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None of the preceding
5
1
Hide problems
P05 [Geometry] - Turkish NMO 1st Round - 2014
Let
D
D
D
be a point on side
[
B
C
]
[BC]
[
BC
]
of
△
A
B
C
\triangle ABC
△
A
BC
such that
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
,
∣
B
D
∣
=
6
|BD|=6
∣
B
D
∣
=
6
and
∣
D
C
∣
=
10
|DC|=10
∣
D
C
∣
=
10
. If the incircles of
△
A
B
D
\triangle ABD
△
A
B
D
and
△
A
D
C
\triangle ADC
△
A
D
C
touch side
[
A
D
]
[AD]
[
A
D
]
at
E
E
E
and
F
F
F
, respectively, what is
∣
E
F
∣
|EF|
∣
EF
∣
?
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2
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3
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9
8
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2
<span class='latex-bold'>(A)</span>\ \dfrac{1}{\sqrt{2}} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{2}{\sqrt{3}} \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{9}{8} \qquad<span class='latex-bold'>(E)</span>\ 2
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2
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8
9
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2
4
1
Hide problems
P04 [Combinatorics] - Turkish NMO 1st Round - 2014
What is the probability of having
2
2
2
adjacent white balls or
2
2
2
adjacent blue balls in a random arrangement of
3
3
3
red,
2
2
2
white and
2
2
2
blue balls?
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2
5
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(
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)
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3
7
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16
35
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10
21
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5
14
<span class='latex-bold'>(A)</span>\ \dfrac{2}{5} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{7} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{16}{35} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{10}{21} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{5}{14}
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2
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7
3
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16
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21
10
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14
5
3
1
Hide problems
P03 [Algebra] - Turkish NMO 1st Round - 2014
For how many integers
n
n
n
, there are four distinct real numbers satisfying the equation
∣
x
2
−
4
x
−
7
∣
=
n
|x^2-4x-7|=n
∣
x
2
−
4
x
−
7∣
=
n
?
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(
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)
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12
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(
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)
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10
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8
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7
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5
<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 10 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 5
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12
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(
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)
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8
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7
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2
1
Hide problems
P02 [Number Theory] - Turkish NMO 1st Round - 2014
How many pairs of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
are there such that
m
n
+
n
+
14
=
(
m
−
1
)
2
mn+n+14=\left (m-1 \right)^2
mn
+
n
+
14
=
(
m
−
1
)
2
?
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a
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16
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12
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8
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6
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2
<span class='latex-bold'>a)</span>\ 16 \qquad<span class='latex-bold'>b)</span>\ 12 \qquad<span class='latex-bold'>c)</span>\ 8 \qquad<span class='latex-bold'>d)</span>\ 6 \qquad<span class='latex-bold'>e)</span>\ 2
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16
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b
)
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12
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8
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d
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6
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2
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2014
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
m
(
D
A
B
^
)
=
m
(
C
B
D
^
)
=
12
0
∘
m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}
m
(
D
A
B
)
=
m
(
CB
D
)
=
12
0
∘
,
∣
A
B
∣
=
2
|AB|=2
∣
A
B
∣
=
2
,
∣
A
D
∣
=
4
|AD|=4
∣
A
D
∣
=
4
and
∣
B
C
∣
=
∣
B
D
∣
|BC|=|BD|
∣
BC
∣
=
∣
B
D
∣
. If the line through
C
C
C
which is parallel to
A
B
AB
A
B
meets
A
D
AD
A
D
at
E
E
E
, what is
∣
C
E
∣
|CE|
∣
CE
∣
?
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(
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8
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(
B
)
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7
<
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c
l
a
s
s
=
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a
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x
−
b
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′
>
(
C
)
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6
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
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5
<
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a
s
s
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(
E
)
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None of the preceding
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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(
A
)
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8
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=
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a
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x
−
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>
(
B
)
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7
<
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p
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c
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a
ss
=
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a
t
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x
−
b
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d
′
>
(
C
)
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6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
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>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
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>
None of the preceding