MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2009 National Olympiad First Round
2009 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
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Flights between 100 cities
There are one-way flights between
100
100
100
cities of a country. It is possible to fly starting from the capital city and visiting all other
99
99
99
cities and returning again to the capital city. Let
N
N
N
be the smallest number of flights inorder to form such a flight combination. Among all flight combinations (satisfying previous condtions),
N
N
N
can be at most ?
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<span class='latex-bold'>(A)</span>\ 1850 \qquad<span class='latex-bold'>(B)</span>\ 2100 \qquad<span class='latex-bold'>(C)</span>\ 2550 \qquad<span class='latex-bold'>(D)</span>\ 3060 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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35
1
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When a_2 a_3 ... a_k > 3
For every
n
≥
2
n \ge 2
n
≥
2
, a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n. What is the least value of positive integer
k
k
k
satisfying
a
2
a
3
⋯
a
k
>
3
a_2a_3\cdots a_k > 3
a
2
a
3
⋯
a
k
>
3
?
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<span class='latex-bold'>(A)</span>\ 100 \qquad<span class='latex-bold'>(B)</span>\ 102 \qquad<span class='latex-bold'>(C)</span>\ 104 \qquad<span class='latex-bold'>(D)</span>\ 106 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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34
1
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min of (x + y^2)(x^2 - y)/(xy)
x
x
x
and
y
y
y
are two distinct positive integers. What is the minimum positive integer value of (x \plus{} y^2)(x^2 \minus{} y)/(xy) ?
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<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ 17
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33
1
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Medians, Centroid, Concylic
A
L
AL
A
L
,
B
M
BM
BM
, and
C
N
CN
CN
are the medians of
△
A
B
C
\triangle ABC
△
A
BC
.
K
K
K
is the intersection of medians. If
C
,
K
,
L
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M
C,K,L,M
C
,
K
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L
,
M
are concyclic and AB \equal{} \sqrt 3, then the median
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N
CN
CN
= ?
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<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ \frac {3\sqrt3}{2} \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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32
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n set with 4 elements
There are
n
n
n
sets having
4
4
4
elements each. The difference set of any two of the sets is equal to one of the
n
n
n
sets.
n
n
n
can be at most ? (A difference set of
A
A
A
and
B
B
B
is
(
A
∖
B
)
∪
(
B
∖
A
)
(A\setminus B)\cup(B\setminus A)
(
A
∖
B
)
∪
(
B
∖
A
)
)
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<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>\ 15 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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31
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|x^3 + 3x^2 - 33x - 3| >= 2x^2
For all
∣
x
∣
≥
n
|x| \ge n
∣
x
∣
≥
n
, the inequality |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2 holds. Integer
n
n
n
can be at least ?
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<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 5
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30
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Is a^2 + b^2 + c^2 a perfect square?
How many of 11^2 \plus{} 13^2 \plus{} 17^2, 24^2 \plus{} 25^2 \plus{} 26^2, 12^2 \plus{} 24^2 \plus{} 36^2, 11^2 \plus{} 12^2 \plus{} 132^2 are perfect square ?
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<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 d)1 \qquad<span class='latex-bold'>(E)</span>\ 0
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29
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Hide problems
Cyclic quadrilateral
P
P
P
is the intersection point of diagonals of cyclic
A
B
C
D
ABCD
A
BC
D
. The circumcenters of
△
A
P
B
\triangle APB
△
A
PB
and
△
C
P
D
\triangle CPD
△
CP
D
lie on circumcircle of
A
B
C
D
ABCD
A
BC
D
. If AC \plus{} BD \equal{} 18, then area of
A
B
C
D
ABCD
A
BC
D
is ?
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<span class='latex-bold'>(A)</span>\ 36 \qquad<span class='latex-bold'>(B)</span>\ \frac {81}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac {36\sqrt 3}{2} \qquad<span class='latex-bold'>(D)</span>\ \frac {81\sqrt 3}{4} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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28
1
Hide problems
Divide Z into n subset no difference of two is prime
We divide entire
Z
Z
Z
into
n
n
n
subsets such that difference of any two elements in a subset will not be a prime number.
n
n
n
is at least ?
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<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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27
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f(1/2009) + f(2/2009) + .. f(2009/2009)
f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}. \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?
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<span class='latex-bold'>(A)</span>\ 1000 \qquad<span class='latex-bold'>(B)</span>\ 1005 \qquad<span class='latex-bold'>(C)</span>\ 1010 \qquad<span class='latex-bold'>(D)</span>\ 2009 \qquad<span class='latex-bold'>(E)</span>\ 2010
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26
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a_0 + 2a_1 + 2^2a_2+ ... + 2^{17}a_{17} = 2^10
For every
0
≤
i
≤
17
0 \le i \le 17
0
≤
i
≤
17
, a_i \equal{} \{ \minus{} 1, 0, 1\}. How many
(
a
0
,
a
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,
…
,
a
17
)
(a_0,a_1, \dots , a_{17})
(
a
0
,
a
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,
…
,
a
17
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18 \minus{}tuples are there satisfying : a_0 \plus{} 2a_1 \plus{} 2^2a_2 \plus{} \cdots \plus{} 2^{17}a_{17} \equal{} 2^{10}
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<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 1
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Incircle touch
The incircle of
△
A
B
C
\triangle ABC
△
A
BC
touches
B
C
BC
BC
,
A
C
AC
A
C
, and
A
B
AB
A
B
at
A
1
A_1
A
1
,
B
1
B_1
B
1
, and
C
1
C_1
C
1
, respectively. The line
A
A
1
AA_1
A
A
1
intersects the incircle at
Q
Q
Q
, again.
A
1
C
1
A_1C_1
A
1
C
1
and
A
1
B
1
A_1B_1
A
1
B
1
intersect the line, passing through
A
A
A
and parallel to
B
C
BC
BC
, at
P
P
P
and
R
R
R
, respectively. If \angle PQC_1 \equal{} 45^\circ and \angle RQB_1 \equal{} 65^\circ, then
∠
P
Q
R
\angle PQR
∠
PQR
will be ?
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∘
<span class='latex-bold'>(A)</span>\ 110^\circ \qquad<span class='latex-bold'>(B)</span>\ 115^\circ \qquad<span class='latex-bold'>(C)</span>\ 120^\circ \qquad<span class='latex-bold'>(D)</span>\ 125^\circ \qquad<span class='latex-bold'>(E)</span>\ 130^\circ
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∘
24
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red and blue rectangles
In xy \minus{}plane, there are
b
b
b
blue and
r
r
r
red rectangles whose sides are parallel to the axis. Any parallel line to the axis can intersect at most one rectangle with same color. For any two rectangle with different colors, there is a line which is parallel to the axis and which intersects only these two rectangles.
(
b
,
r
)
(b,r)
(
b
,
r
)
cannot be ?
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<span class='latex-bold'>(A)</span>\ (1,7) \qquad<span class='latex-bold'>(B)</span>\ (2,6) \qquad<span class='latex-bold'>(C)</span>\ (3,4) \qquad<span class='latex-bold'>(D)</span>\ (3,3) \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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23
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Hide problems
Min [x(x + 4)(x + 8)(x + 12)]
The minimum value of x(x \plus{} 4)(x \plus{} 8)(x \plus{} 12) in real numbers is ?
(A)
\ \minus{} 240 \qquad
(B)
\ \minus{} 252 \qquad
(C)
\ \minus{} 256 \qquad
(D)
\ \minus{} 260 \qquad
(E)
\ \minus{} 280
22
1
Hide problems
a{n+1} = a{n}^3 + a{n}^2 and mod11
(a_n)_{n \equal{} 0}^\infty is a sequence on integers. For every
n
≥
0
n \ge 0
n
≥
0
, a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2. The number of distinct residues of
a
i
a_i
a
i
in
(
m
o
d
11
)
\pmod {11}
(
mod
11
)
can be at most?
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<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6
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21
1
Hide problems
P inside ABC with 70, 10, 10, 40, x, 50 - x
AB \equal{} AC, \angle BAC \equal{} 80^\circ. Let
E
E
E
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
such that AE \equal{} EC and \angle EAC \equal{} 10^\circ. What is the measure of
∠
E
B
C
\angle EBC
∠
EBC
?
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<span class='latex-bold'>(A)</span>\ 10^\circ \qquad<span class='latex-bold'>(B)</span>\ 15^\circ \qquad<span class='latex-bold'>(C)</span>\ 20^\circ \qquad<span class='latex-bold'>(D)</span>\ 25^\circ \qquad<span class='latex-bold'>(E)</span>\ 30^\circ
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20
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5-digit numbers with even number of digits are even.
Let
A
A
A
be the numbers of 5-digit positive numbers satisfying following condition: The first digit is odd. Remaining
0
0
0
, or
2
2
2
or
4
4
4
digit/digits are even.Let
B
B
B
be the numbers of 5-digit positive numbers satisfying following condition: The first digit is even. Remaining
0
0
0
, or
2
2
2
or
4
4
4
digit/digits are even. A \minus{} B \equal{} ?
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<span class='latex-bold'>(A)</span>\ 5000 \qquad<span class='latex-bold'>(B)</span>\ 4640 \qquad<span class='latex-bold'>(C)</span>\ 3200 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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19
1
Hide problems
roots of two quadratic equation
a
a
a
is a real number.
x
1
x_1
x
1
and
x
2
x_2
x
2
are the distinct roots of x^2 \plus{} ax \plus{} 2 \equal{} x.
x
3
x_3
x
3
and
x
4
x_4
x
4
are the distinct roots of (x \minus{} a)^2 \plus{} a(x \minus{} a) \plus{} 2 \equal{} x. If x_3 \minus{} x_1 \equal{} 3(x_4 \minus{} x_2), then x_4 \minus{} x_2 will be ?
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<span class='latex-bold'>(A)</span>\ \frac {a}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac {a}{3} \qquad<span class='latex-bold'>(C)</span>\ \frac {2a}{3} \qquad<span class='latex-bold'>(D)</span>\ \frac {3a}{2} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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1
Hide problems
n^3 = 1 mod 455
1
≤
n
≤
455
1 \le n \le 455
1
≤
n
≤
455
and
n
3
≡
1
(
m
o
d
455
)
n^3 \equiv 1 \pmod {455}
n
3
≡
1
(
mod
455
)
. The number of solutions is ?
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<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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Equilateral ABC, AD=8, BD = 13, DC=?
A
B
C
ABC
A
BC
is an equilateral triangle.
D
D
D
is a point inside
△
A
B
C
\triangle ABC
△
A
BC
such that AD \equal{} 8, BD \equal{} 13, and \angle ADC \equal{} 120^\circ. What is the length of
D
C
DC
D
C
?
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<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 13 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ 16
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16
1
Hide problems
x+19y = 0 mod 23 and x+y < 69
x \plus{} 19y \equiv 0 \pmod {23} and x \plus{} y < 69. How many pairs of
(
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(x,y)
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are there in positive integers?
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<span class='latex-bold'>(A)</span>\ 100 \qquad<span class='latex-bold'>(B)</span>\ 102 \qquad<span class='latex-bold'>(C)</span>\ 105 \qquad<span class='latex-bold'>(D)</span>\ 109 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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15
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|x| + |y| = 13 and x^2 + 7x - 3y + y^2 = ?
For real numbers, if |x| \plus{} |y| \equal{} 13, then x^2 \plus{} 7x \minus{} 3y \plus{} y^2 cannot be
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<span class='latex-bold'>(A)</span>\ 208 \qquad<span class='latex-bold'>(B)</span>\ 15\sqrt {2} \qquad<span class='latex-bold'>(C)</span>\ \frac {35}{2} \qquad<span class='latex-bold'>(D)</span>\ 37 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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14
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mn | (2008 x 2009 x 2010)
For how many ordered pairs of positive integers
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(m,n)
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,
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divides
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2008 \cdot 2009 \cdot 2010
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?
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<span class='latex-bold'>(A)</span>\ 2\cdot3^7\cdot 5 \qquad<span class='latex-bold'>(B)</span>\ 2^5\cdot3\cdot 5 \qquad<span class='latex-bold'>(C)</span>\ 2^5\cdot3^7\cdot 5 \qquad<span class='latex-bold'>(D)</span>\ 2^3\cdot3^5\cdot 5^2 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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trapezoid inequality
In trapezoid
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ABCD
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BC
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AB \parallel CD
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<
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\angle CAB < 90^\circ
∠
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, AB \equal{} 5, CD \equal{} 3, AC \equal{} 15. What are the sum of different integer values of possible
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BD
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?
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<span class='latex-bold'>(A)</span>\ 101 \qquad<span class='latex-bold'>(B)</span>\ 108 \qquad<span class='latex-bold'>(C)</span>\ 115 \qquad<span class='latex-bold'>(D)</span>\ 125 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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12
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8-digit number with only two are same
How many 8-digit numbers are there such that exactly 7 digits are all different?
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<span class='latex-bold'>(A)</span>\ {{9}\choose{3}}^2 \cdot 6! \cdot 3 \qquad<span class='latex-bold'>(B)</span>\ {{8}\choose{3}}^2 \cdot 7! \qquad<span class='latex-bold'>(C)</span>\ {{8}\choose{3}}^2 \cdot 7! \cdot 3 \\ \qquad<span class='latex-bold'>(D)</span>\ {{7}\choose{3}}^2 \cdot 7! \qquad<span class='latex-bold'>(E)</span>\ {{9}\choose{4}}^2 \cdot 6! \cdot 8
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11
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a{n} a{n + 3} = a{n + 2} a{n + 5}
(a_n)_{n \equal{} 1}^\infty is defined on real numbers with a_n \not \equal{} 0, a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5} and a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6. So a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?
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<span class='latex-bold'>(A)</span>\ 21 \qquad<span class='latex-bold'>(B)</span>\ 42 \qquad<span class='latex-bold'>(C)</span>\ 63 \qquad<span class='latex-bold'>(D)</span>\ 882 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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10
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n^4 + 4n^3 + 3n^2 - 2n + 7 is prime
For how many integer
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n
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, P \equal{} n^4 \plus{} 4n^3 \plus{} 3n^2 \minus{} 2n \plus{} 7 is prime?
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C
)
<
/
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a
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>
3
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(
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)
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4
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(
E
)
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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>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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1
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)
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3
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4
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(
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Infinitely many
9
1
Hide problems
Inradius of triangles formed by diagonals
Let
E
E
E
be the intersection of the diagonals of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. The perimeters of
△
A
E
B
\triangle AEB
△
A
EB
,
△
B
E
C
\triangle BEC
△
BEC
,
△
C
E
D
\triangle CED
△
CE
D
, and
△
D
E
A
\triangle DEA
△
D
E
A
are all same. If inradii of
△
A
E
B
\triangle AEB
△
A
EB
,
△
B
E
C
\triangle BEC
△
BEC
,
△
C
E
D
\triangle CED
△
CE
D
are
3
,
4
,
6
3,4,6
3
,
4
,
6
, respectively, then inradius of
△
D
E
A
\triangle DEA
△
D
E
A
will be ?
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9
2
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(
B
)
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7
2
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(
C
)
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13
3
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(
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)
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5
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None
<span class='latex-bold'>(A)</span>\ \frac {9}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac {7}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac {13}{3} \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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2
9
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2
7
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3
13
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(
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5
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None
8
1
Hide problems
two elements whose sum is a perfect square
S \equal{} \{1,2,\dots,n\} is divided into two subsets. How the set is divided, if there exist two elements whose sum is a perfect square, then
n
n
n
is at least ?
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13
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)
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14
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15
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)
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16
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17
<span class='latex-bold'>(A)</span>\ 13 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 15 \qquad<span class='latex-bold'>(D)</span>\ 16 \qquad<span class='latex-bold'>(E)</span>\ 17
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13
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(
B
)
<
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14
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)
<
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15
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)
<
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16
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)
<
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17
7
1
Hide problems
uncommon real roots of the two polynomials
The product of uncommon real roots of the two polynomials x^4 \plus{} 2x^3 \minus{} 8x^2 \minus{} 6x \plus{} 15 and x^3 \plus{} 4x^2 \minus{} x \minus{} 10 is ?
(A)
\ \minus{} 4 \qquad
(B)
\ 4 \qquad
(C)
\ \minus{} 6 \qquad
(D)
\ 6 \qquad
(E)
\ \text{None}
6
1
Hide problems
a^2b + ab^2 = 2009201020092010
How many ordered integer pairs of
(
a
,
b
)
(a,b)
(
a
,
b
)
satisfying a^2b \plus{} ab^2 \equal{} 2009201020092010 ?
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None
<span class='latex-bold'>(A)</span> 4 \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{None}
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(
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)
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(
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(
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)
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0
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(
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None
5
1
Hide problems
exradius of hypotenuse
What is the perimeter of the right triangle whose exradius of the hypotenuse is
30
30
30
?
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(
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)
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40
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45
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50
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(
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)
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60
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75
<span class='latex-bold'>(A)</span>\ 40 \qquad<span class='latex-bold'>(B)</span>\ 45 \qquad<span class='latex-bold'>(C)</span>\ 50 \qquad<span class='latex-bold'>(D)</span>\ 60 \qquad<span class='latex-bold'>(E)</span>\ 75
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)
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45
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50
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60
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75
4
1
Hide problems
n = 7a + 5b
Let
a
,
b
a,b
a
,
b
be integers greater than
1
1
1
. What is the largest
n
n
n
which cannot be written in the form n \equal{} 7a \plus{} 5b ?
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82
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47
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45
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)
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42
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None
<span class='latex-bold'>(A)</span>\ 82 \qquad<span class='latex-bold'>(B)</span>\ 47 \qquad<span class='latex-bold'>(C)</span>\ 45 \qquad<span class='latex-bold'>(D)</span>\ 42 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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82
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(
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)
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47
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45
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)
<
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42
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(
E
)
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None
3
1
Hide problems
x = \sqrt[3]{11 + \sqrt{337}} + \sqrt[3]{11 - \sqrt{337}}
If x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}, then x^3 \plus{} 18x = ?
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)
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24
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)
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22
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(
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20
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(
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11
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10
<span class='latex-bold'>(A)</span>\ 24 \qquad<span class='latex-bold'>(B)</span>\ 22 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ 10
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24
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(
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)
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22
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(
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20
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10
2
1
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a^2 + b^4 = 5^n
If
a
,
b
,
n
a,b,n
a
,
b
,
n
are positive integers, number of solutions of the equaition a^2 \plus{} b^4 \equal{} 5^n is
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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>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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Infinitely many
1
1
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Interior point
A
B
C
D
ABCD
A
BC
D
is a square. Let
E
E
E
be a point on the segment
B
C
BC
BC
and
F
F
F
be a point on the segment
E
D
ED
E
D
. If DF \equal{} BF and EF \equal{} BE, then
∠
D
F
A
\angle DFA
∠
D
F
A
is
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∘
<span class='latex-bold'>(A)</span>\ 45^\circ \qquad<span class='latex-bold'>(B)</span>\ 60^\circ \qquad<span class='latex-bold'>(C)</span>\ 75^\circ \qquad<span class='latex-bold'>(D)</span>\ 80^\circ \qquad<span class='latex-bold'>(E)</span>\ 85^\circ
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