Problem 1
What is the value of![\[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]](https://latex.artofproblemsolving.com/6/7/6/67606c2b8141058216ca18ee5e71821458b74b01.png)

![\[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]](https://latex.artofproblemsolving.com/6/7/6/67606c2b8141058216ca18ee5e71821458b74b01.png)

Problem 2
What is the hundreds digit of 


Problem 3
Ana and Bonita were born on the same date in different years,
years apart. Last year Ana was
times as old as Bonita. This year Ana's age is the square of Bonita's age. What is 




Problem 4
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 5
What is the greatest number of consecutive integers whose sum is 


Problem 6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
- a square
- a rectangle that is not a square
- a rhombus that is not a square
- a parallelogram that is not a rectangle or a rhombus
- an isosceles trapezoid that is not a parallelogram

Problem 7
Two lines with slopes
and
intersect at
. What is the area of the triangle enclosed by these two lines and the line 





Problem 8
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 9
What is the greatest three-digit positive integer
for which the sum of the first
positive integers is
a divisor of the product of the first
positive integers?





Problem 10
A rectangular floor that is
feet wide and
feet long is tiled with
one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?




Problem 11
How many positive integer divisors of
are perfect squares or perfect cubes (or both)?


Problem 12
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 13
Let
be an isosceles triangle with
and
. Construct the circle with diameter
, and let
and
be the other intersection points of the circle with the sides
and
, respectively. Let
be the intersection of the diagonals of the quadrilateral
. What is the degree measure of 












Problem 14
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 15
A sequence of numbers is defined recursively by
,
, and
for all
. Then
can be written as
, where
and
are relatively prime positive integers. 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 16
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 17
A child builds towers using identically shaped cubes of different color. How many different towers with a height
cubes can the child build with
red cubes,
blue cubes, and
green cubes? (One cube will be left out.)





Problem 18
For some positive integer
, the repeating base-
representation of the (base-ten) fraction
is
. What is
?






Problem 19
What is the least possible value of
where
is a real number?
![\[(x+1)(x+2)(x+3)(x+4)+2019\]](https://latex.artofproblemsolving.com/1/2/8/128007ee3b97f3ff741a1e668139edf8c13ba7e1.png)


Problem 20
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 21
A sphere with center
has radius 6. 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 22
Real numbers between 0 and 1, 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 0 if the second flip is heads and 1 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 23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number
, then Todd must say the next two numbers (
and
), then Tucker must say the next three numbers (
,
,
), then Tadd must say the next four numbers (
,
,
,
), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number
is reached. What is the
th number said by Tadd?













Problem 24
Let
,
, and
be the distinct roots of the polynomial
. It is given that there exist real numbers
,
, and
such that
for all
. What is
?







![\[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]](https://latex.artofproblemsolving.com/e/d/4/ed44e7481e867b959ea84d896b30b222fe5efea3.png)



Problem 25
For how many integers
between
and
, inclusive, is
an integer? (Recall that
.)



![\[\frac{(n^2-1)!}{(n!)^{n}}\]](https://latex.artofproblemsolving.com/c/0/9/c09ec519b3d2598b4d89348c3999cd0e715c9f8b.png)


AoPS
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