Problem 1
Alicia had two containers. The first was
full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was
full of water. What is the ratio of the volume of the first container to the volume of the second container?



Problem 2
Consider the statement, "If
is not prime, then
is prime." Which of the following values of
is a counterexample to this statement?




Problem 3
Which one of the following rigid transformations (isometries) maps the line segment
onto the line segment
so that the image of
is
and the image of
is 















Problem 4
A positive integer
satisfies the equation
. What is the sum of the digits of
?




Problem 5
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or
pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of
?



Problem 6
In a given plane, points
and
are
units apart. How many points
are there in the plane such that the perimeter of
is
units and the area of
is
square units?









Problem 7
What is the sum of all real numbers
for which the median of the numbers
and
is equal to the mean of those five numbers?




Problem 8
Let
. What is the value of the sum




Problem 9
For how many integral values of
can a triangle of positive area be formed having side lengths
?



Problem 10
The figure below is a map showing
cities and
roads connecting certain pairs of cities. Paula wishes to travel along exactly
of those roads, starting at city
and ending at city
without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)





![[asy] import olympiad; unitsize(50); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { pair A = (j,i); dot(A); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { if (j != 3) { draw((j,i)--(j+1,i)); } if (i != 2) { draw((j,i)--(j,i+1)); } } } label("$A$", (0,2), W); label("$L$", (3,0), E); [/asy]](https://latex.artofproblemsolving.com/2/6/8/268cd67d45f5b55a849d4cc611db555eb4e89633.png)
How many different routes can Paula take?

Problem 11
How many unordered pairs of edges of a given cube determine a plane?

Problem 12
Right triangle
with right angle at
is constructed outwards on the hypotenuse
of isosceles right triangle
with leg length
, as shown, so that the two triangles have equal perimeters. What is
?![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(8.016233639805293cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145; /* image dimensions */ draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.)); draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.)); /* draw figures */ draw((-2.,3.)--(-2.,-1.), linewidth(2.)); draw((-2.,-1.)--(2.,-1.), linewidth(2.)); draw((2.,-1.)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.)); label("$D$",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14)); label("$A$",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14)); label("$B$",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14)); label("$C$",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14)); label("$1$",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14)); label("$1$",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,3.),linewidth(4.pt) + dotstyle); dot((-2.,-1.),linewidth(4.pt) + dotstyle); dot((2.,-1.),linewidth(4.pt) + dotstyle); dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]](https://latex.artofproblemsolving.com/e/9/1/e9154dcee5b662add15f00ec3aa10f8e25240c0d.png)






![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(8.016233639805293cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145; /* image dimensions */ draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.)); draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.)); /* draw figures */ draw((-2.,3.)--(-2.,-1.), linewidth(2.)); draw((-2.,-1.)--(2.,-1.), linewidth(2.)); draw((2.,-1.)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.)); label("$D$",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14)); label("$A$",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14)); label("$B$",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14)); label("$C$",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14)); label("$1$",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14)); label("$1$",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,3.),linewidth(4.pt) + dotstyle); dot((-2.,-1.),linewidth(4.pt) + dotstyle); dot((2.,-1.),linewidth(4.pt) + dotstyle); dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]](https://latex.artofproblemsolving.com/e/9/1/e9154dcee5b662add15f00ec3aa10f8e25240c0d.png)

Problem 13
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin
is
for
What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?




Problem 14
Let
be the set of all positive integer divisors of
How many numbers are the product of two distinct elements of 




Problem 15
As shown in the figure, line segment
is trisected by points
and
so that
Three semicircles of radius
and
have their diameters on
and are tangent to line
at
and
respectively. A circle of radius
has its center on
The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
where
and
are positive integers and
and
are relatively prime. What is
?













![\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]](https://latex.artofproblemsolving.com/d/8/b/d8bdd0e53bdd2b5b50aea9510be4bd6c45e2b867.png)





![[asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy]](https://latex.artofproblemsolving.com/b/e/5/be5538c17cf2af3d37027f4996d5a8ae5cd4aef1.png)

Problem 16
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a
chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?


Problem 17
How many nonzero complex numbers
have the property that
and
when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?




Problem 18
Square pyramid
has base
which measures
cm on a side, and altitude
perpendicular to the base
which measures
cm. Point
lies on
one third of the way from
to
point
lies on
one third of the way from
to
and point
lies on
two thirds of the way from
to
What is the area, in square centimeters, of 




















Problem 19
Raashan, Sylvia, and Ted play the following game. Each starts with
. A bell rings every
seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives
to that player. What is the probability that after the bell has rung
times, each player will have
? (For example, Raashan and Ted may each decide to give
to Sylvia, and Sylvia may decide to give her her dollar to Ted, at which point Raashan will have
, Sylvia will have
, and Ted will have
, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their
to, and the holdings will be the same at the end of the second round.)











Problem 20
Points
and
lie on circle
in the plane. Suppose that the tangent lines to
at
and
intersect at a point on the
-axis. What is the area of
?









Problem 21
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is
and the roots are
and
then the requirement is that
.)





Problem 22
Define a sequence recursively by
and
for all nonnegative integers
Let
be the least positive integer such that
In which of the following intervals does
lie?

![\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]](https://latex.artofproblemsolving.com/9/3/4/934a32343352988b8d8328a7ef93c7553a1ce642.png)


![\[x_m\leq 4+\frac{1}{2^{20}}.\]](https://latex.artofproblemsolving.com/e/8/8/e889e3b4b1d11c0fd3ce0eda55858624351b69fd.png)

![$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$](https://latex.artofproblemsolving.com/6/c/2/6c2f679a426a4923a52cc6f0de03e28c75165307.png)
Problem 23
How many sequences of
s and
s of length
are there that begin with a
, end with a
, contain no two consecutive
s, and contain no three consecutive
s?








Problem 24
Let
Let
denote all points in the complex plane of the form
where
and
What is the area of
?







Problem 25
Let
be a convex quadrilateral with
and
Suppose that the centroids of
and
form the vertices of an equilateral triangle. What is the maximum possible value of the area of
?







AoPS
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