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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
2005 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
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P36 [Combinatorics] - Turkish NMO 1st Round - 2005
n
n
n
wrestlers participate in a tournament such that any two wrestlers wrestle exactly once. After a match, the winner gets
2
2
2
points, the loser gets no point, and each wrestlers gets
1
1
1
point if a tie occurs. After the tournament finishes, the wrestler with highest points is the wrestler with lowest number of wins. What is the least value of
n
n
n
?
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9
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ 9
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9
32
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P32 [Combinatorics] - Turkish NMO 1st Round - 2005
Ali chooses one of the stones from a group of
2005
2005
2005
stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone?
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<span class='latex-bold'>(A)</span>\ 11 \qquad<span class='latex-bold'>(B)</span>\ 13 \qquad<span class='latex-bold'>(C)</span>\ 17 \qquad<span class='latex-bold'>(D)</span>\ 18 \qquad<span class='latex-bold'>(E)</span>\ 19
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P28 [Combinatorics] - Turkish NMO 1st Round - 2005
How many solutions does the equation
a
!
=
b
!
c
!
a ! = b ! c !
a
!
=
b
!
c
!
have where
a
a
a
,
b
b
b
,
c
c
c
are integers greater than
1
1
1
?
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Infinitely many
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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Infinitely many
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P24 [Combinatorics] - Turkish NMO 1st Round - 2005
There are
20
20
20
people in a certain community.
10
10
10
of them speak English,
10
10
10
of them speak German, and
10
10
10
of them speak French. We call a committee to a
3
3
3
-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
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1140
<span class='latex-bold'>(A)</span>\ 120 \qquad<span class='latex-bold'>(B)</span>\ 380 \qquad<span class='latex-bold'>(C)</span>\ 570 \qquad<span class='latex-bold'>(D)</span>\ 1020 \qquad<span class='latex-bold'>(E)</span>\ 1140
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1140
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P20 [Combinatorics] - Turkish NMO 1st Round - 2005
We are swapping two different digits of a number in each step. If we start with the number
12345
12345
12345
, which of the following cannot be got after an even number of steps?
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13425
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35142
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53124
<span class='latex-bold'>(A)</span>\ 13425 \qquad<span class='latex-bold'>(B)</span>\ 21435 \qquad<span class='latex-bold'>(C)</span>\ 35142 \qquad<span class='latex-bold'>(D)</span>\ 43125 \qquad<span class='latex-bold'>(E)</span>\ 53124
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35142
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43125
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53124
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P16 [Combinatorics] - Turkish NMO 1st Round - 2005
100
100
100
stones, each weighs
1
1
1
kg or
10
10
10
kgs or
50
50
50
kgs, weighs
500
500
500
kgs in total. How many values can the number of stones weighing
10
10
10
kgs take?
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<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>\ 4
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P12 [Combinatorics] - Turkish NMO 1st Round - 2005
Ali and Veli goes to hunting. The probability that each will successfully hit a duck is
1
/
2
1/2
1/2
on any given shot. During the hunt, Ali shoots
12
12
12
times, and Veli shoots
13
13
13
times. What is the probability that Veli hits more ducks than Ali?
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3
4
<span class='latex-bold'>(A)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(B)</span>\ \dfrac{13}{25} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{13}{24} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{7}{13} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{3}{4}
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13
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13
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7
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3
8
1
Hide problems
P08 [Combinatorics] - Turkish NMO 1st Round - 2005
How many natural number triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
are there such that
x
y
z
=
1
0
6
xyz = 10^6
x
yz
=
1
0
6
?
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568
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784
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812
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816
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824
<span class='latex-bold'>(A)</span>\ 568 \qquad<span class='latex-bold'>(B)</span>\ 784 \qquad<span class='latex-bold'>(C)</span>\ 812 \qquad<span class='latex-bold'>(D)</span>\ 816 \qquad<span class='latex-bold'>(E)</span>\ 824
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812
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816
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824
4
1
Hide problems
P04 [Combinatorics] - Turkish NMO 1st Round - 2005
How many
6
6
6
-digit positive integers whose digits are different from
0
0
0
are there such that each number generated by rearranging the digits of the original number is always divisible by
7
7
7
?
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11
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77
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255
<span class='latex-bold'>(A)</span>\ 11 \qquad<span class='latex-bold'>(B)</span>\ 77 \qquad<span class='latex-bold'>(C)</span>\ 133 \qquad<span class='latex-bold'>(D)</span>\ 166 \qquad<span class='latex-bold'>(E)</span>\ 255
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255
35
1
Hide problems
P35 [Algebra] - Turkish NMO 1st Round - 2005
If for every real
x
x
x
,
a
x
2
+
b
x
+
c
≥
0
ax^2 + bx+c \geq 0
a
x
2
+
b
x
+
c
≥
0
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are reals such that
a
<
b
a<b
a
<
b
, what is the smallest value of
a
+
b
+
c
b
−
a
\dfrac{a+b+c}{b-a}
b
−
a
a
+
b
+
c
?
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5
3
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2
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7
2
<span class='latex-bold'>(A)</span>\ \dfrac{5}{\sqrt 3} \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ \dfrac{\sqrt 5}2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \dfrac{\sqrt 7}2
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2
7
31
1
Hide problems
P31 [Algebra] - Turkish NMO 1st Round - 2005
If the equation system
f
(
x
)
+
g
(
x
)
=
0
f
(
x
)
−
(
g
(
x
)
)
3
=
0
\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}
f
(
x
)
+
g
(
x
)
f
(
x
)
−
(
g
(
x
)
)
3
=
=
0
0
has more than one real roots, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are reals and
f
(
x
)
=
x
2
+
a
x
+
b
f(x)=x^2 + ax+b
f
(
x
)
=
x
2
+
a
x
+
b
,
g
(
x
)
=
x
2
+
c
x
+
d
g(x)=x^2 + cx + d
g
(
x
)
=
x
2
+
c
x
+
d
, at most how many distinct real roots can the equation
f
(
x
)
g
(
x
)
=
0
f(x)g(x) = 0
f
(
x
)
g
(
x
)
=
0
have?
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4
<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>\ 4
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27
1
Hide problems
P27 [Algebra] - Turkish NMO 1st Round - 2005
What is the maximum value of the difference between the largest real root and the smallest real root of the equation system
a
x
2
+
b
x
+
c
=
0
b
x
2
+
c
x
+
a
=
0
c
x
2
+
a
x
+
b
=
0
\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}
a
x
2
+
b
x
+
c
b
x
2
+
c
x
+
a
c
x
2
+
a
x
+
b
=
=
=
0
0
0
, where at least one of the reals
a
,
b
,
c
a,b,c
a
,
b
,
c
is non-zero?
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2
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There is no upper-bound
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \text{There is no upper-bound}
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2
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There is no upper-bound
23
1
Hide problems
P23 [Algebra] - Turkish NMO 1st Round - 2005
How many solutions does the equation system
x
−
1
x
y
−
3
=
3
−
x
−
y
7
−
x
2
−
y
2
=
y
−
2
x
y
−
4
\dfrac{x-1}{xy-3}=\dfrac{3-x-y}{7-x^2-y^2} = \dfrac{y-2}{xy-4}
x
y
−
3
x
−
1
=
7
−
x
2
−
y
2
3
−
x
−
y
=
x
y
−
4
y
−
2
have?
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A
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0
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>
(
B
)
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−
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C
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a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
<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>\ 4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
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
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
19
1
Hide problems
P19 [Algebra] - Turkish NMO 1st Round - 2005
What is the greatest real root of the equation
x
3
−
x
2
−
x
−
1
3
=
0
x^3-x^2-x-\frac 13 = 0
x
3
−
x
2
−
x
−
3
1
=
0
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
−
2
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
3
−
2
3
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
3
3
−
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
4
3
−
1
<
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>\ \dfrac{\sqrt {3} - \sqrt{2}}{2} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{\sqrt [3]{3} - \sqrt[3]{2}}{2} \qquad<span class='latex-bold'>(C)</span>\ \dfrac 1{\sqrt[3] {3} - 1} \qquad<span class='latex-bold'>(D)</span>\ \dfrac 1{\sqrt[3] {4} - 1} \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
3
−
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
3
3
−
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
3
−
1
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
4
−
1
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
15
1
Hide problems
P15 [Algebra] - Turkish NMO 1st Round - 2005
For how many positive real numbers
a
a
a
has the equation
a
2
x
2
+
a
x
+
1
−
7
a
2
=
0
a^2x^2 + ax+1-7a^2 = 0
a
2
x
2
+
a
x
+
1
−
7
a
2
=
0
two distinct integer roots?
<
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
>
Infinitely many
<
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>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
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
>
Infinitely many
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
11
1
Hide problems
P11 [Algebra] - Turkish NMO 1st Round - 2005
For the real pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying the equation
x
2
+
y
2
+
2
x
−
6
y
=
6
x^2 + y^2 + 2x - 6y = 6
x
2
+
y
2
+
2
x
−
6
y
=
6
, which of the following cannot be equal to
(
x
−
1
)
2
+
(
y
−
2
)
2
(x-1)^2 + (y-2)^2
(
x
−
1
)
2
+
(
y
−
2
)
2
?
<
s
p
a
n
c
l
a
s
s
=
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l
a
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x
−
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>
(
A
)
<
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a
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>
2
<
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
>
16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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a
n
>
23
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
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>
30
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 9 \qquad<span class='latex-bold'>(C)</span>\ 16 \qquad<span class='latex-bold'>(D)</span>\ 23 \qquad<span class='latex-bold'>(E)</span>\ 30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
<
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
>
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
23
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
30
7
1
Hide problems
P07 [Algebra] - Turkish NMO 1st Round - 2005
What is the greatest value of
sin
x
cos
y
+
sin
y
cos
z
+
sin
z
cos
x
\sin x \cos y + \sin y \cos z + \sin z \cos x
sin
x
cos
y
+
sin
y
cos
z
+
sin
z
cos
x
, where
x
,
y
,
z
x,y,z
x
,
y
,
z
are real numbers?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
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
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
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
<span class='latex-bold'>(A)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3}2 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ 3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
3
3
1
Hide problems
P03 [Algebra] - Turkish NMO 1st Round - 2005
If the difference between the greatest and the smallest root of the equation
x
3
−
6
x
2
+
5
x^3 - 6x^2 + 5
x
3
−
6
x
2
+
5
is equal to
F
F
F
, which of the following is true?
(A)
\ 0 \leq F < 2
(B)
\ 2 \leq F < 4
(C)
\ 4 \leq F < 6
(D)
\ 6 \leq F < 8
(E)
\ 8 \leq F
34
1
Hide problems
P34 [Number Theory] - Turkish NMO 1st Round - 2005
How many triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers are there such that
x
y
z
=
510510
xyz=510510
x
yz
=
510510
and
x
2
y
+
y
2
z
+
z
2
x
=
x
y
2
+
y
z
2
+
z
x
2
x^2y+y^2z+z^2x = xy^2 + yz^2 + zx^2
x
2
y
+
y
2
z
+
z
2
x
=
x
y
2
+
y
z
2
+
z
x
2
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
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
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
8
<
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>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
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
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
30
1
Hide problems
P30 [Number Theory] - Turkish NMO 1st Round - 2005
Which of the following cannot be the number of positive integer divisors of the number
n
2
+
1
n^2+1
n
2
+
1
, where
n
n
n
is an integer?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
8
<
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>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
26
1
Hide problems
P26 [Number Theory] - Turkish NMO 1st Round - 2005
For every positive integer
n
n
n
,
f
(
2
n
+
1
)
=
2
f
(
2
n
)
f(2n+1)=2f(2n)
f
(
2
n
+
1
)
=
2
f
(
2
n
)
,
f
(
2
n
)
=
f
(
2
n
−
1
)
+
1
f(2n)=f(2n-1)+1
f
(
2
n
)
=
f
(
2
n
−
1
)
+
1
, and
f
(
1
)
=
0
f(1)=0
f
(
1
)
=
0
. What is the remainder when
f
(
2005
)
f(2005)
f
(
2005
)
is divided by
5
5
5
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
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a
n
>
0
<
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
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
<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>\ 4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
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
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
22
1
Hide problems
P22 [Number Theory] - Turkish NMO 1st Round - 2005
For which
k
k
k
, there is no integer pair
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
2
−
y
2
=
k
x^2 - y^2 = k
x
2
−
y
2
=
k
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2005
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2006
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2007
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2008
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2009
<span class='latex-bold'>(A)</span>\ 2005 \qquad<span class='latex-bold'>(B)</span>\ 2006 \qquad<span class='latex-bold'>(C)</span>\ 2007 \qquad<span class='latex-bold'>(D)</span>\ 2008 \qquad<span class='latex-bold'>(E)</span>\ 2009
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2005
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2006
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2007
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2008
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2009
18
1
Hide problems
P18 [Number Theory] - Turkish NMO 1st Round - 2005
How many integers
0
≤
x
<
121
0\leq x < 121
0
≤
x
<
121
are there such that
x
5
+
5
x
2
+
x
+
1
≡
0
(
m
o
d
121
)
x^5+5x^2 + x + 1 \equiv 0 \pmod{121}
x
5
+
5
x
2
+
x
+
1
≡
0
(
mod
121
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
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
<
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
>
5
<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>\ 5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
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
>
2
<
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
>
5
14
1
Hide problems
P14 [Number Theory] - Turkish NMO 1st Round - 2005
We call a number
1
0
3
<
n
<
1
0
6
10^3 < n < 10^6
1
0
3
<
n
<
1
0
6
a balanced number if the sum of its last three digits is equal to the sum of its other digits. What is the sum of all balanced numbers in
m
o
d
13
\bmod {13}
mod
13
?
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
7
<
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
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a
n
>
12
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 7 \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
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
7
<
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
10
1
Hide problems
P10 [Number Theory] - Turkish NMO 1st Round - 2005
Which of the following does not divide
n
2225
−
n
2005
n^{2225}-n^{2005}
n
2225
−
n
2005
for every integer value of
n
n
n
?
<
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
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
7
<
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
>
23
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ 23
<
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
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
7
<
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
>
23
6
1
Hide problems
P06 [Number Theory] - Turkish NMO 1st Round - 2005
Which of the following divides
3
3
n
+
1
+
5
3
n
+
2
+
7
3
n
+
3
3^{3n+1} + 5^{3n+2}+7^{3n+3}
3
3
n
+
1
+
5
3
n
+
2
+
7
3
n
+
3
for every positive integer
n
n
n
?
<
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
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
7
<
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
>
53
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ 53
<
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
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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an
>
11
<
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p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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an
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53
2
1
Hide problems
P02 [Number Theory] - Turkish NMO 1st Round - 2005
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
be positive integers such that none of them is a multiple of
5
5
5
. What is the largest integer
n
<
2005
n<2005
n
<
2005
, such that
a
1
4
+
a
2
4
+
⋯
+
a
n
4
a_1^4 + a_2^4 + \cdots + a_n^4
a
1
4
+
a
2
4
+
⋯
+
a
n
4
is divisible by
5
5
5
?
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c
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s
s
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−
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(
A
)
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a
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>
2000
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
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>
2001
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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a
n
>
2002
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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a
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>
2003
<
s
p
a
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c
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a
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s
=
′
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−
b
o
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>
(
E
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<
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2004
<span class='latex-bold'>(A)</span>\ 2000 \qquad<span class='latex-bold'>(B)</span>\ 2001 \qquad<span class='latex-bold'>(C)</span>\ 2002 \qquad<span class='latex-bold'>(D)</span>\ 2003 \qquad<span class='latex-bold'>(E)</span>\ 2004
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a
t
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−
b
o
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d
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(
A
)
<
/
s
p
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>
2000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2001
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2002
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2003
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2004
33
1
Hide problems
P33 [Geometry] - Turkish NMO 1st Round - 2005
Let
K
K
K
be the intersection of diagonals of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, where
∣
A
B
∣
=
∣
B
C
∣
|AB|=|BC|
∣
A
B
∣
=
∣
BC
∣
,
∣
B
K
∣
=
b
|BK|=b
∣
B
K
∣
=
b
, and
∣
D
K
∣
=
d
|DK|=d
∣
DK
∣
=
d
. What is
∣
A
B
∣
|AB|
∣
A
B
∣
?
<
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p
a
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l
a
s
s
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a
t
e
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−
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o
l
d
′
>
(
A
)
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a
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>
d
2
+
b
d
<
s
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a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
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a
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>
b
2
+
b
d
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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a
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>
2
b
d
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
(
b
2
+
d
2
−
b
d
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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a
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>
b
d
<span class='latex-bold'>(A)</span>\ \sqrt{d^2 + bd} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{b^2+bd} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{2bd} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{2(b^2+d^2-bd)} \qquad<span class='latex-bold'>(E)</span>\ \sqrt{bd}
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c
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a
ss
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′
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a
t
e
x
−
b
o
l
d
′
>
(
A
)
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/
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p
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>
d
2
+
b
d
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
b
2
+
b
d
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
b
d
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
(
b
2
+
d
2
−
b
d
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
b
d
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2005
Let
h
1
h_1
h
1
and
h
2
h_2
h
2
be the altitudes of a triangle drawn to the sides with length
5
5
5
and
2
6
2\sqrt 6
2
6
, respectively. If
5
+
h
1
≤
2
6
+
h
2
5+h_1 \leq 2\sqrt 6 + h_2
5
+
h
1
≤
2
6
+
h
2
, what is the third side of the triangle?
<
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s
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−
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o
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(
A
)
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n
>
5
<
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c
l
a
s
s
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′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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p
a
n
>
2
6
<
s
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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a
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>
3
6
<
s
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
5
3
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 2\sqrt 6 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 6 \qquad<span class='latex-bold'>(E)</span>\ 5\sqrt 3
<
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p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
5
3
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2005
Let
E
E
E
,
F
F
F
,
G
G
G
be points on sides
[
A
B
]
[AB]
[
A
B
]
,
[
B
C
]
[BC]
[
BC
]
,
[
C
D
]
[CD]
[
C
D
]
of the rectangle
A
B
C
D
ABCD
A
BC
D
, respectively, such that
∣
B
F
∣
=
∣
F
Q
∣
|BF|=|FQ|
∣
BF
∣
=
∣
FQ
∣
,
m
(
F
G
E
^
)
=
9
0
∘
m(\widehat{FGE})=90^\circ
m
(
FGE
)
=
9
0
∘
,
∣
B
C
∣
=
4
3
/
5
|BC|=4\sqrt 3 / 5
∣
BC
∣
=
4
3
/5
, and
∣
E
F
∣
=
5
|EF|=\sqrt 5
∣
EF
∣
=
5
. What is
∣
B
F
∣
|BF|
∣
BF
∣
?
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
10
−
2
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
−
1
<
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
>
11
−
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
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>
1
<span class='latex-bold'>(A)</span>\ \dfrac{\sqrt{10} - \sqrt{2}}{2} \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 -1 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{\sqrt{11} - \sqrt{3}}{2} \qquad<span class='latex-bold'>(E)</span>\ 1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
10
−
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
−
1
<
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
>
2
11
−
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2005
What is the radius of the circle passing through the center of the square
A
B
C
D
ABCD
A
BC
D
with side length
1
1
1
, its corner
A
A
A
, and midpoint of its side
[
B
C
]
[BC]
[
BC
]
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
10
4
<span class='latex-bold'>(A)</span>\ \dfrac {\sqrt 3}4 \qquad<span class='latex-bold'>(B)</span>\ \dfrac {\sqrt 5}4 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ \dfrac {\sqrt {10}}4
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
10
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2005
Construct outer squares
A
B
M
N
ABMN
A
BMN
,
B
C
K
L
BCKL
BC
K
L
,
A
C
P
Q
ACPQ
A
CPQ
on sides
[
A
B
]
[AB]
[
A
B
]
,
[
B
C
]
[BC]
[
BC
]
,
[
C
A
]
[CA]
[
C
A
]
of triangle
A
B
C
ABC
A
BC
, respectively. Construct squares
N
Q
Z
T
NQZT
NQZT
and
K
P
Y
X
KPYX
K
P
Y
X
on segments
[
N
Q
]
[NQ]
[
NQ
]
and
[
K
P
]
[KP]
[
K
P
]
. If
A
r
e
a
(
A
B
M
N
)
−
A
r
e
a
(
B
C
K
L
)
=
1
Area(ABMN) - Area(BCKL)=1
A
re
a
(
A
BMN
)
−
A
re
a
(
BC
K
L
)
=
1
, what is
A
r
e
a
(
N
Q
Z
T
)
−
A
r
e
a
(
K
P
Y
X
)
Area(NQZT)-Area(KPYX)
A
re
a
(
NQZT
)
−
A
re
a
(
K
P
Y
X
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
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<span class='latex-bold'>(A)</span>\ \dfrac 34 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 53 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4
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4
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2005
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid such that its diagonal is
3
\sqrt 3
3
and its base angle is
6
0
∘
60^\circ
6
0
∘
, where
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
. Let
P
P
P
be a point on the plane of the trapezoid such that
∣
P
A
∣
=
1
|PA|=1
∣
P
A
∣
=
1
and
∣
P
D
∣
=
3
|PD|=3
∣
P
D
∣
=
3
. Which of the following can be the length of
[
P
C
]
[PC]
[
PC
]
?
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6
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B
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2
2
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s
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(
C
)
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>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
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>
3
3
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E
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7
<span class='latex-bold'>(A)</span>\ \sqrt 6 \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 2 \sqrt 3 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 7
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6
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a
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=
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a
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x
−
b
o
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d
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>
(
B
)
<
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>
2
2
<
s
p
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l
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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>
(
C
)
<
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>
2
3
<
s
p
an
c
l
a
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=
′
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a
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x
−
b
o
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d
′
>
(
D
)
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>
3
3
<
s
p
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a
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=
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a
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x
−
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>
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E
)
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>
7
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2005
Let
A
B
C
ABC
A
BC
be a triangle with circumradius
1
1
1
. If the center of the circle passing through
A
A
A
,
C
C
C
, and the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
lies on the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
, what is
∣
A
C
∣
|AC|
∣
A
C
∣
?
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A
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−
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>
(
B
)
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3
<
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c
l
a
s
s
=
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a
t
e
x
−
b
o
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d
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>
(
C
)
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a
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>
3
2
<
s
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c
l
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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>
2
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E
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3
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 3
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A
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2
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x
−
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o
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>
(
B
)
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3
<
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p
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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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d
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>
(
C
)
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2
3
<
s
p
an
c
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a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
D
)
<
/
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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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x
−
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>
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E
)
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3
5
1
Hide problems
P05 [Geometry] - Turkish NMO 1st Round - 2005
Let
M
M
M
be the intersection of diagonals of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, where
m
(
A
M
B
^
)
=
6
0
∘
m(\widehat{AMB})=60^\circ
m
(
A
MB
)
=
6
0
∘
. Let the points
O
1
O_1
O
1
,
O
2
O_2
O
2
,
O
3
O_3
O
3
,
O
4
O_4
O
4
be the circumcenters of the triangles
A
B
M
ABM
A
BM
,
B
C
M
BCM
BCM
,
C
D
M
CDM
C
D
M
,
D
A
M
DAM
D
A
M
, respectively. What is
A
r
e
a
(
A
B
C
D
)
/
A
r
e
a
(
O
1
O
2
O
3
O
4
)
Area(ABCD)/Area(O_1O_2O_3O_4)
A
re
a
(
A
BC
D
)
/
A
re
a
(
O
1
O
2
O
3
O
4
)
?
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2
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B
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3
2
<
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−
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>
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C
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2
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x
−
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>
(
D
)
<
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>
1
+
2
3
2
<
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<span class='latex-bold'>(A)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3}2 \qquad<span class='latex-bold'>(D)</span>\ \dfrac {1+2\sqrt 3}2 \qquad<span class='latex-bold'>(E)</span>\ \dfrac {1+\sqrt 3}2
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1
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−
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B
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3
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p
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c
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a
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x
−
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C
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2
3
<
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p
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a
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=
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a
t
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x
−
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o
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d
′
>
(
D
)
<
/
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p
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>
2
1
+
2
3
<
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p
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a
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=
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a
t
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x
−
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d
′
>
(
E
)
<
/
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>
2
1
+
3
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2005
The circle with radius
4
4
4
passing through
A
A
A
and
B
B
B
is tangent to the circle with radius
3
3
3
passing through
A
A
A
and
C
C
C
, where
∣
A
B
∣
=
2
|AB|=2
∣
A
B
∣
=
2
. If the line
B
C
BC
BC
is tangent to the second circle, what is
∣
B
C
∣
|BC|
∣
BC
∣
?
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7
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(
B
)
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2
+
43
2
<
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a
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s
=
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x
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(
C
)
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5
2
<
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x
−
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>
(
D
)
<
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4
+
9
<
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7
<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 2 + \dfrac{\sqrt{43}}2 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 52 \qquad<span class='latex-bold'>(D)</span>\ 4 + \sqrt 9 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 7
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7
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x
−
b
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(
B
)
<
/
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>
2
+
2
43
<
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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
′
>
(
C
)
<
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an
>
2
5
<
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p
an
c
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a
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=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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an
>
4
+
9
<
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c
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a
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=
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−
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o
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d
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(
E
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7