Problem 1
The area of a pizza with radius
is
percent larger than the area of a pizza with radius
inches. What is the integer closest to
?





Problem 2
Suppose
is
of
. What percent of
is
?






Problem 3
A box contains
red balls,
green balls,
yellow balls,
blue balls,
white balls, and
black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least
balls of a single color will be drawn?








Problem 4
What is the greatest number of consecutive integers whose sum is
?


Problem 5
Two lines with slopes
and
intersect at
. What is the area of the triangle enclosed by these two lines and the line
?





Problem 6
The figure below shows line
with a regular, infinite, recurring pattern of squares and line segments.

![[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]](https://latex.artofproblemsolving.com/a/2/f/a2f165e74d6ddc1b021c8af4c74374a1a78d2815.png)
- some rotation around a point of line
- some translation in the direction parallel to line
- the reflection across line
- some reflection across a line perpendicular to line

Problem 7
Melanie computes the mean
, the median
, and the modes of the
values that are the dates in the months of
. Thus her data consist of
,
, . . . ,
,
,
, and
. Let
be the median of the modes. Which of the following statements is true?


















Problem 8
For a set of four distinct lines in a plane, there are exactly
distinct points that lie on two or more of the lines. What is the sum of all possible values of
?



Problem 9
A sequence of numbers is defined recursively by
,
, and
for all
. Then
can be written as
, where
and
are relatively prime positive inegers. What is 


![\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]](https://latex.artofproblemsolving.com/4/c/6/4c6c8899cddbad7fd227142de526641de7c68e5b.png)







Problem 10
The figure below shows
circles of radius
within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius
?



![[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]](https://latex.artofproblemsolving.com/f/c/4/fc485e1337895925d85375cce1a9d3908ca8b7e3.png)

Problem 11
For some positive integer
, the repeating base-
representation of the (base-ten) fraction
is
. What is
?






Problem 12
Positive real numbers
and
satisfy
and
. What is
?






Problem 13
How many ways are there to paint each of the integers
either red, green, or blue so that each number has a different color from each of its proper divisors?


Problem 14
For a certain complex number
, the polynomial
has exactly 4 distinct roots. What is
?

![\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]](https://latex.artofproblemsolving.com/0/7/8/07858e6ad5c6c6f79e5c145fe6e55e5cee64606e.png)


Problem 15
Positive real numbers
and
have the property that![\[\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100\]](https://latex.artofproblemsolving.com/3/9/0/390863333afc338a0d3a3157a3e2be1d552a68c2.png)


![\[\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100\]](https://latex.artofproblemsolving.com/3/9/0/390863333afc338a0d3a3157a3e2be1d552a68c2.png)
and all four terms on the left are positive integers, where
denotes the base-
logarithm. What is
?




Problem 16
The numbers
are randomly placed into the
squares of a
grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?




Problem 17
Let
denote the sum of the
th powers of the roots of the polynomial
. In particular,
,
, and
. Let
,
, and
be real numbers such that
for
,
,
What is
?















Problem 18
A sphere with center
has radius
. A triangle with sides of length
and
is situated in space so that each of its sides is tangent to the sphere. What is the distance between
and the plane determined by the triangle?






Problem 19
In
with integer side lengths,
What is the least possible perimeter for
?

![\[\cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}.\]](https://latex.artofproblemsolving.com/3/3/c/33c5cf6fca4eabad219869260b59af5b93d6e5a2.png)


Problem 20
Real numbers between
and
, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is
if the second flip is heads and
if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval
. Two random numbers
and
are chosen independently in this manner. What is the probability that
?




![$[0,1]$](https://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)




Problem 21
Let
What is![\[(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?\]](https://latex.artofproblemsolving.com/f/7/8/f78b9af129c3d9d63e4b6dde4d253f43de5b1f43.png)
![\[z=\frac{1+i}{\sqrt{2}}.\]](https://latex.artofproblemsolving.com/b/b/e/bbe83cc2920853600cd4b1481848779dfc774102.png)
![\[(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?\]](https://latex.artofproblemsolving.com/f/7/8/f78b9af129c3d9d63e4b6dde4d253f43de5b1f43.png)

Problem 22
Circles
and
, both centered at
, have radii
and
, respectively. Equilateral triangle
, whose interior lies in the interior of
but in the exterior of
, has vertex
on
, and the line containing side
is tangent to
. Segments
and
intersect at
, and
. Then
can be written in the form
for positive integers
,
,
,
with
. What is
? 


























Problem 23
Define binary operations
and
by
for all real numbers
and
for which these expressions are defined. The sequence
is defined recursively by
and
for all integers
. To the nearest integer, what is
?


![\[a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}\]](https://latex.artofproblemsolving.com/5/f/0/5f0ef0b94787e769e936ea5ec8657b108b356c9e.png)




![\[a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}\]](https://latex.artofproblemsolving.com/f/1/b/f1b2d6d1783ff5064ff088b0b0ff162c3de58efc.png)



Problem 24
For how many integers
between
and
, inclusive, is
an integer? (Recall that
.)



![\[\frac{(n^2-1)!}{(n!)^n}\]](https://latex.artofproblemsolving.com/2/f/b/2fb0a7fcc7e8ea99426bae70e641eac53ea8d7e6.png)


Problem 25
Let
be a triangle whose angle measures are exactly
,
, and
. For each positive integer
define
to be the foot of the altitude from
to line
. Likewise, define
to be the foot of the altitude from
to line
, and
to be the foot of the altitude from
to line
. What is the least positive integer
for which
is obtuse?

















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