MathDB

2009 Purple Comet Problems

Part of Purple Comet Problems

Subcontests

(25)
24
1

Purple Comet 2009 HS Problem 24

A right circular cone pointing downward forms an angle of 60∘60^\circ at its vertex. Sphere SS with radius 11 is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere SS, and to the side of the cone. The radius of these congruent spheres can be written as a+bc\tfrac{a+\sqrt{b}}{c} where aa, bb, and cc are positive integers such that aa and cc are relatively prime. Find a+b+ca + b + c.
[asy] size(150); real t=0.12; void ball(pair x, real r, real h, bool ww=true) { pair xx=yscale(t)*x+(0,h); path P=circle(xx,r); unfill(P); draw(P); if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r)); } pair X=(0,0); real H=17, h=5, R=h/2; draw(H*dir(120)--(0,0)--H*dir(60)); draw(ellipse((0,0.87*H),H/2,t*H/2)); pair Y=(R,h+2*R),C=(0,h); real r; for(int k=0;k<20;++k) { r=-(dir(30)*Y).x; Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3; } ball(Y.x*dir(90),r,Y.y,false); ball(X,R,h); ball(Y.x*dir(-30),r,Y.y); ball(Y.x*dir(210),r,Y.y);[/asy]
23
1

Purple Comet 2009 HS Problem 23

Square ABCDABCD has side length 44. Points EE and FF are the midpoints of sides ABAB and CDCD, respectively. Eight 11 by 22 rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers mm and nn so that mn\tfrac{m}{n} is the probability that none of the rectangles crosses the line segment EFEF (as in the arrangement on the right). Find m+nm + n.
[asy] size(200); defaultpen(linewidth(0.8)+fontsize(10pt)); real r = 7; path square=origin--(4,0)--(4,4)--(0,4)--cycle; draw(square^^shift((r,0))*square,linewidth(1)); draw((1,4)--(1,0)^^(3,4)--(3,0)^^(0,2)--(1,2)^^(1,3)--(3,3)^^(1,1)--(3,1)^^(2,3)--(2,1)^^(3,2)--(4,2)); draw(shift((r,0))*((2,4)--(2,0)^^(0,2)--(4,2)^^(0,1)--(4,1)^^(0,3)--(2,3)^^(3,4)--(3,2))); label("A",(4,4),NE); label("A",(4+r,4),NE); label("B",(0,4),NW); label("B",(r,4),NW); label("C",(0,0),SW); label("C",(r,0),SW); label("D",(4,0),SE); label("D",(4+r,0),SE); label("E",(2,4),N); label("E",(2+r,4),N); label("F",(2,0),S); label("F",(2+r,0),S); [/asy]
22
1

Purple Comet 2009 HS Problem 22

The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is a+bc\tfrac{a+\sqrt{b}}{c} where aa, bb, and cc are positive integers, and aa and cc are relatively prime. Find a+b+ca + b + c.
[asy] size(200); real f(real x) {return 1.2*exp(2/3*log(16-x^2));} path Q=graph(f,-3.99999,3.99999); path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle}; for(int k=0;k<3;++k) { fill(P[k],grey); draw(P[k]); } draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]
21
1

Purple Comet 2009 HS Problem 21

A cylinder radius 1212 and a cylinder radius 3636 are held tangent to each other with a tight band. The length of the band is mk+nπm\sqrt{k}+n\pi where mm, kk, and nn are positive integers, and kk is not divisible by the square of any prime. Find m+k+nm + k + n.
[asy] size(150); real t=0.3; void cyl(pair x, real r, real h) { pair xx=(x.x,t*x.y); path B=ellipse(xx,r,t*r), T=ellipse((x.x,t*x.y+h),r,t*r), S=xx+(r,0)--xx+(r,h)--(xx+(-r,h))--xx-(r,0); unfill(S--cycle); draw(S); unfill(B); draw(B); unfill(T); draw(T); } real h=8, R=3,r=1.2; pair X=(0,0), Y=(R+r)*dir(-50); cyl(X,R,h); draw(shift((0,5))*yscale(t)*arc(X,R,180,360)); cyl(Y,r,h); void str (pair x, pair y, real R, real r, real h, real w) { real u=(angle(y-x)+asin((R-r)/(R+r)))*180/pi+270; path P=yscale(t)*(arc(x,R,180,u)--arc(y,r,u,360)); path Q=shift((0,h))*P--shift((0,h+w))*reverse(P)--cycle; fill(Q,grey);draw(Q); } str(X,Y,R,r,3.5,1.5);[/asy]
7
2
6
2
25
1
19
1
20
1
18
1
17
1
16
1
15
2

Purple Comet 2009 HS Problem 15

What is the remainder when 7897^{8^9} is divided by 1000?1000?

Purple Comet 2009 MS Problem 15

We have twenty-seven 11 by 11 cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one 33 by 33 cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of all the 11 by 11 cubes. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,7)--(-1,4)); draw((-1,9.15)--(-3.42,8.21)); draw((-1,9.15)--(1.42,8.21)); draw((-1,7)--(1.42,8.21)); draw((1.42,7.21)--(-1,6)); draw((1.42,6.21)--(-1,5)); draw((1.42,5.21)--(-1,4)); draw((1.42,8.21)--(1.42,5.21)); draw((-3.42,8.21)--(-3.42,5.21)); draw((-3.42,7.21)--(-1,6)); draw((-3.42,8.21)--(-1,7)); draw((-1,4)--(-3.42,5.21)); draw((-3.42,6.21)--(-1,5)); draw((-2.61,7.8)--(-2.61,4.8)); draw((-1.8,4.4)--(-1.8,7.4)); draw((-0.2,7.4)--(-0.2,4.4)); draw((0.61,4.8)--(0.61,7.8)); label("2",(-1.07,9.01),SE*labelscalefactor); label("9",(-1.88,8.65),SE*labelscalefactor); label("1",(-2.68,8.33),SE*labelscalefactor); label("3",(-0.38,8.72),SE*labelscalefactor); draw((-1.8,7.4)--(0.63,8.52)); draw((-0.27,8.87)--(-2.61,7.8)); draw((-2.65,8.51)--(-0.2,7.4)); draw((-1.77,8.85)--(0.61,7.8)); label("7",(-1.12,8.33),SE*labelscalefactor); label("5",(-1.9,7.91),SE*labelscalefactor); label("1",(0.58,8.33),SE*labelscalefactor); label("18",(-0.36,7.89),SE*labelscalefactor); label("1",(-1.07,7.55),SE*labelscalefactor); label("1",(-0.66,6.89),SE*labelscalefactor); label("5",(-0.68,5.8),SE*labelscalefactor); label("1",(-0.68,4.83),SE*labelscalefactor); label("2",(0.09,7.27),SE*labelscalefactor); label("1",(0.15,6.24),SE*labelscalefactor); label("2",(0.11,5.26),SE*labelscalefactor); label("1",(0.89,7.61),SE*labelscalefactor); label("3",(0.89,6.63),SE*labelscalefactor); label("9",(0.92,5.62),SE*labelscalefactor); label("18",(-3.18,7.63),SE*labelscalefactor); label("2",(-3.07,6.61),SE*labelscalefactor); label("2",(-3.09,5.62),SE*labelscalefactor); label("1",(-2.29,7.25),SE*labelscalefactor); label("3",(-2.27,6.22),SE*labelscalefactor); label("5",(-2.29,5.2),SE*labelscalefactor); label("7",(-1.49,6.89),SE*labelscalefactor); label("34",(-1.52,5.81),SE*labelscalefactor); label("1",(-1.41,4.86),SE*labelscalefactor); [/asy]
14
2

Purple Comet 2009 HS Problem 14

Let ABCDABCD be a trapezoid with ABAB parallel to CD,ABCD, AB has length 1,1, and CDCD has length 41.41. Let points XX and YY lie on sides ADAD and BC,BC, respectively, such that XYXY is parallel to ABAB and CD,CD, and XYXY has length 31.31. Let mm and nn be two relatively prime positive integers such that the ratio of the area of ABYXABYX to the area of CDXYCDXY is mn.\tfrac{m}{n}. Find m+2n.m+2n.

Purple Comet 2009 MS Problem 14

Rectangle ABCDABCD measures 7070 by 4040. Eighteen points (including AA and CC) are marked on the diagonal ACAC dividing the diagonal into 1717 congruent pieces. Twenty-two points (including A and B) are marked on the side ABAB dividing the side into 2121 congruent pieces. Seventeen non-overlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left 1717 of the 2121 congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these 1717 triangles.
[asy] size(200); defaultpen(linewidth(0.8)); pair A=origin,B=(21,0),C=(21,12),D=(0,12); path P=origin; draw(A--B--C--D--cycle--C); for (int r = 1; r <= 17;++r) { P=P--(21*r/17,12*r/17)--(r,0); } P=P--cycle; filldraw(P,gray(0.7)); label("AA",A,SW); label("BB",B,SE); label("CC",C,NE); label("DD",D,NW); [/asy]
13
2
12
2
11
2
10
2

Purple Comet 2009 HS Problem 10

Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.

Purple Comet 2009 MS Problem 10

The diagram shows a 2020 by 2020 square ABCDABCD. The points EE, FF, and GG are equally spaced on side BCBC. The points HH, II, JJ, and KK on side DADA are placed so that the triangles BKEBKE, EJFEJF, FIGFIG, and GHCGHC are isosceles. Points LL and MM are midpoints of the sides ABAB and CDCD, respectively. Find the total area of the shaded regions.
[asy] size(175); defaultpen(linewidth(0.8)); real r=20/8; pair x[]; draw(origin--(0,20)--(20,20)--(20,0)--cycle); string label[]={"B","K","E","J","F","I","G","H","C"}; for(int i=1;i<=7;i=i+1) { if(floor(i/2)==i/2) { x=(i*r,0); label(""+label+""+label+"",x,S); } else { x=(i*r,20); label(""+label+""+label+"",x,N); } } filldraw(origin--x[1]--x[2]--x[3]--x[4]--x[5]--x[6]--x[7]--(20,0)--(20,10)--(0,10)--cycle,gray); label("BB",origin,SW); label("CC",(20,0),SE); label("AA",(0,20),NW); label("DD",(20,20),NE); label("MM",(20,10),E); label("LL",(0,10),W); [/asy]
9
2
8
2
1
2
5
2
4
2

Purple Comet 2009 HS Problem 4

There are three bags of marbles. Bag two has twice as many marbles as bag one. Bag three has three times as many marbles as bag one. Half the marbles in bag one, one third the marbles in bag two, and one fourth the marbles in bag three are green. If all three bags of marbles are dumped into a single pile, mn\frac{m}{n} of the marbles in the pile would be green where mm and nn are relatively prime positive integers. Find m+n.m + n.

Purple Comet 2009 MS Problem 4

John, Paul, George, and Ringo baked a circular pie. Each cut a piece that was a sector of the circle. John took one-third of the whole pie. Paul took one-fourth of the whole pie. George took one-fifth of the whole pie. Ringo took one-sixth of the whole pie. At the end the pie had one sector remaining. Find the measure in degrees of the angle formed by this remaining sector.
3
2

Purple Comet 2009 HS Problem 3

In the diagram ABCDEFGABCDEFG is a regular heptagon (a 77 sided polygon). Shown is the star AEBFCGDAEBFCGD. The degree measure of the obtuse angle formed by AEAE and CGCG is mn\dfrac{m}{n} where m and n are relatively prime positive integers. Find m+nm + n.
[asy] size(150); defaultpen(linewidth(1)); string lab[]={"A","B","C","D","E","F","G"}; real r = 360/7; pair A=dir(90-r),B=dir(90),C=dir(90+r),D=dir(90+2*r),E=dir(90+3*r),F=dir(90+4*r),G=dir(90+5*r); draw(A--E--B--F--C--G--D--cycle); for(int k = -1;k <= 5;++k) { label(""+lab[k+1]+""+lab[k+1]+"",dir(90+k*r),dir(90+k*r)); } [/asy]

Purple Comet 2009 MS Question 3

The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is 27+28+29+30+1+2+3=120.27 + 28 + 29 + 30 + 1 + 2 + 3 = 120. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
2
2