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math contest problems competition high school

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1. A ball is dropped from a height of 5 ft on a surface that causes it tobounce up 10% of the distance that it fell. If the probability that the ball ison its way up at any point in time is represented by the fraction (
m/n
) inlowest terms, where
m
and
n
are positive integers, what is
m
+
n
?2. A circle on the coordinate plane is centered at the srcin and has radius
r
. What proportion of the circle is located below the line
y
=
r/
2?3. A recursive sequence has terms
t
n
in the form
a
n
b
n
and is deﬁned such thatall even numbered terms are the reciprocal of the immediately preceding termand in all odd numbered terms,
a
n
and
b
n
are equal to
a
n
−
2
+1 and
b
n
−
2
+1respectively. If
t
1
=
23
what is:(
a
2005
+
a
2006
+
a
2007
+
a
2008
)(
b
2005
+
b
2006
+
b
2007
+
b
2008
)4. Isosceles triangle
ABC
is inscribed in a circle with radius
r
1
such thatside
AB
is a diameter of the circle. If a circle inscribed within triangle
ABC
has radius
r
2
, what is
r
2
r
1
?5. Let
F
,
T
, and
W
be digits with (100
F
+ 10
T
+
W
)(
F
+
T
+
W
) = 2008.What is
F
·
T
·
W
?6. How many positive 3 digit integers satisfy the property that the tens digitis the geometric mean of the hundreds and units digits?7. Let
E
and
F
be the respective midpoints of
BC
and
CD
on unit square
ABCD
. If lines
DE
and
BF
intersect at
G
, what is the area of triangle
BDG
?1
8. Let
n
represent the number of 16 digit palindromes divisible by 11. Whatis the remainder when
n
is divided by 11?9. What is the sum of the tens and units digits of 9
99
−
11
11
?10. If tan
θ >
1 and
∞
n
=1
1tan
n
θ
= sin
θ
+cos
θ
, ﬁnd
θ
in the interval 0
≤
θ
≤
π
11. The function
f
(
x
) =
ax
3
+
bx
2
+
cx
+
d
has 3 positive real roots. If thesum of the roots of
f
(
x
) is 4, what is the largest possible value of
ca
?12. Two circles are located on the coordinate plane such that the distancebetween their centers is
√
2 units. If both circles have a radius of 1 unit,what is the area of the region enclosed by their intersection?13. If
r
is equal to the sum of the ﬁrst
n
positive integers and
s
is equal tothe sum of the ﬁrst
n
positive integer squares, for how many values of
n
inthe interval 0
< n <
2008 will
sr
be an integer?14. If cos2
θ
=
14
, what is sin
4
θ
+ cos
4
θ
?15. What is smallest positive integer
m
for which the expression (
√
3 + 3
i
2)
m
is an integer?16. If the function
f
(
x
) =
x
2
−
bx
+ 9999 has positive integer roots, what isthe diﬀerence between the minimum and maximum possible values of
b
?2
17. If
f
(
x
) = (
x
2
+ 4
x
+ 3)
2
+ 4(
x
2
+ 4
x
+ 3) + 3, what is the sum of itscomplex roots?18. If
x
2
+
y
2
+
z
2
= 22
x
+10
y
+2
z
, what is the maximum possible value of
x
+
y
+
z
?19. A right triangle has perimeter of length 7 and hypotenuse of length 3. If
θ
is the largest non-right angle in the triangle, what is the value of cos
θ
?20. The function
f
(
x
) is a quadratic such that
f
(0) =
−
f
(0). If
g
(
x
) =
f
(
x
)
−
f
(
x
) and the sum of the roots of
g
(
x
) is
54
, ﬁnd the sum of the roots of
f
(
x
).3

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