Found problems: 28
2006 Purple Comet Problems, 21
In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.
2006 Purple Comet Problems, 14
Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs.
2006 Purple Comet Problems, 23
We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.
2006 Purple Comet Problems, 12
We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?
2006 Purple Comet Problems, 1
The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?
2006 Purple Comet Problems, 8
Evaluate $\frac{2i}{\frac{1}{30 - 5i} - \frac{1}{30 + 5i}}$ where $i=\sqrt{-1}$.
2006 Purple Comet Problems, 7
Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction $\tfrac{m}{n}$where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2006 Purple Comet Problems, 14
The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?
2006 Purple Comet Problems, 13
$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?
2006 Purple Comet Problems, 15
A snowman is built on a level plane by placing a ball radius $6$ on top of a ball radius $8$ on top of a ball radius $10$ as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.
[asy]
size(150);
draw(circle((0,0),24));
draw(ellipse((0,0),24,9));
draw(circle((0,-56),32));
draw(ellipse((0,-56),32,12));
draw(circle((0,-128),40));
draw(ellipse((0,-128),40,15));
[/asy]
2006 Purple Comet Problems, 19
There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?
2006 Purple Comet Problems, 10
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy]
2006 Purple Comet Problems, 11
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.
2006 Purple Comet Problems, 4
A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12\%$ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?
2006 Purple Comet Problems, 15
A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$.
[asy]
import three;
size(250);
defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2");
currentprojection = orthographic(0,-15,5);
draw(circle((0,0,0), 15),dashes);
draw(circle((0,0,80), 15));
draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80));
draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes);
draw("48", (-24,0,-20)--(24,0,-20));
draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17));
label("30", (0,0,-15));
draw("60", (50,0,0)--(50,0,60));
draw("20", (50,0,60)--(50,0,80));
draw((50,0,60)--(47,0,60));[/asy]
2006 Purple Comet Problems, 9
How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7?
[asy]
int n;
n=0;
for (int i=0; i<=7;++i)
{
draw((i,0)--(i,7));
draw((0,i)--(7,i));
for (int a=0; a<=7;++a)
{
if ((a != 7)&&(i != 7))
{
n=n+1;
label((string) n,(a,i),(1.5,2));
}
}
}
[/asy]
2006 Purple Comet Problems, 16
$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.
2006 Purple Comet Problems, 24
A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as $\frac{n + k\sqrt{2}}{m}$ where $m$, $n$, and $k$ are positive integers and $m$ and $n$ have no factors in common. Find $m + n + k$.
[asy]
size(200);
pair O=(0,0);
real R=10, r=4.7;
draw(arc(O,R,0,180)--cycle);
pair P=(sqrt((R-r)^2-r^2),r),Q;
draw(circle(P,r));
real a=0,b=r,c;
for(int k=0;k<20;++k)
{
c=(a+b)/2;
Q=(-sqrt((R-c)^2-c^2),c);
if(abs(P-Q)>c+r) a=c; else b=c;
}
draw(circle(Q,c));[/asy]
2006 Purple Comet Problems, 17
A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$.
[asy]
import three;
size(250);
defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2");
currentprojection = orthographic(0,-15,5);
draw(circle((0,0,0), 15),dashes);
draw(circle((0,0,80), 15));
draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80));
draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes);
draw("48", (-24,0,-20)--(24,0,-20));
draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17));
label("30", (0,0,-15));
draw("60", (50,0,0)--(50,0,60));
draw("20", (50,0,60)--(50,0,80));
draw((50,0,60)--(47,0,60));[/asy]
2006 Purple Comet Problems, 18
In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?
2006 Purple Comet Problems, 25
Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.
2006 Purple Comet Problems, 3
Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$. $AN$ and $MB$ intersect at $X$. Given that $\angle AXB$ measures $127^{\circ}$ and the minor arc $AM$ measures $14^{\circ}$, compute the measure of the angle at $P$.
[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label("$A$",A,plain.NE);label("$B$",B,plain.NW);label("$M$",M,SE);
label("$P$",P,E);label("$N$",N,dir(250));label("$X$",X,plain.N);[/asy]
2006 Purple Comet Problems, 20
Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$.
2006 Purple Comet Problems, 2
At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How many dollars would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?
2006 Purple Comet Problems, 22
Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$.