Subcontests
(25)Neat Grid
The numbers 1,2,…,9 are randomly placed into the 9 squares of a 3×3 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?<spanclass=′latex−bold′>(A)</span>1/21<spanclass=′latex−bold′>(B)</span>1/14<spanclass=′latex−bold′>(C)</span>5/63<spanclass=′latex−bold′>(D)</span>2/21<spanclass=′latex−bold′>(E)</span>1/7 Three Semicircle IQ Test
As shown in the figure, line segment AD is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1, \overarc{AEB},\overarc{BFC}, and \overarc{CGD}, have their diameters on AD, and are tangent to line EG at E,F, and G, respectively. A circle of radius 2 has its center on F. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
ba⋅π−c+d,
where a,b,c, and d are positive integers and a and b are relatively prime. What is a+b+c+d?[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]
<spanclass=′latex−bold′>(A)</span>13<spanclass=′latex−bold′>(B)</span>14<spanclass=′latex−bold′>(C)</span>15<spanclass=′latex−bold′>(D)</span>16<spanclass=′latex−bold′>(E)</span>17 Travis = Mr. Beast?
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 1, then Todd must say the next two numbers (2 and 3), then Tucker must say the next three numbers (4, 5, 6), then Tadd must say the next four numbers (7, 8, 9, 10), 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 10,000 is reached. What is the 2019th number said by Tadd?<spanclass=′latex−bold′>(A)</span> 5743<spanclass=′latex−bold′>(B)</span> 5885<spanclass=′latex−bold′>(C)</span> 5979<spanclass=′latex−bold′>(D)</span> 6001<spanclass=′latex−bold′>(E)</span> 6011 Red Ball is Superior
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 k is 2−k for k=1,2,3,…. What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? <spanclass=′latex−bold′>(A)</span>41<spanclass=′latex−bold′>(B)</span>72<spanclass=′latex−bold′>(C)</span>31<spanclass=′latex−bold′>(D)</span>83<spanclass=′latex−bold′>(E)</span>73 Circles in circles
The figure below shows 13 circles of radius 1 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 1?[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]<spanclass=′latex−bold′>(A)</span>4π3<spanclass=′latex−bold′>(B)</span>7π<spanclass=′latex−bold′>(C)</span>π(33+2)<spanclass=′latex−bold′>(D)</span>10π(3−1)<spanclass=′latex−bold′>(E)</span>π(3+6) Median #7 Strikes Back
What is the sum of all real numbers x for which the median of the numbers 4,6,8,17, and x is equal to the mean of those five numbers?<spanclass=′latex−bold′>(A)</span>−5<spanclass=′latex−bold′>(B)</span>0<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>415<spanclass=′latex−bold′>(E)</span>435
Means, Medians, and Modes
Melanie computes the mean μ, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thus her data consist of 12 1s, 12 2s, . . . , 12 28s, 11 29s, 11 30s, and 7 31s. Let d be the median of the modes. Which of the following statements is true?<spanclass=′latex−bold′>(A)</span>μ<d<M<spanclass=′latex−bold′>(B)</span>M<d<μ<spanclass=′latex−bold′>(C)</span>d=M=μ<spanclass=′latex−bold′>(D)</span>d<M<μ<spanclass=′latex−bold′>(E)</span>d<μ<M Range of weird function
The function f is defined by f(x)=⌊∣x∣⌋−⌊x⌋for all real numbers x, where ⌊r⌋ denotes the greatest integer less than or equal to the real number r. What is the range of f?<spanclass=′latex−bold′>(A)</span>{−1,0}<spanclass=′latex−bold′>(B)</span>The set of nonpositive integers<spanclass=′latex−bold′>(C)</span>{−1,0,1} <spanclass=′latex−bold′>(D)</span>{0}<spanclass=′latex−bold′>(E)</span>The set of nonnegative integers Symmetry
The figure below shows line ℓ 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]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
[*] some rotation around a point of line ℓ
[*] some translation in the direction parallel to line ℓ
[*] the reflection across line ℓ
[*] some reflection across a line perpendicular to line ℓ<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>4 Equilaterals and quadrilaterals
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?[asy]
pen white = gray(1);
pen gray = gray(0.5);
draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle);
fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray);
draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle);
fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white);
draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle);
fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white);
draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle);
fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white);
draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle);
fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white);
[/asy]<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>12−43<spanclass=′latex−bold′>(C)</span>33<spanclass=′latex−bold′>(D)</span>43<spanclass=′latex−bold′>(E)</span>16−3 Partial Fraction Decomposition
Let p, q, and r be the distinct roots of the polynomial x3−22x2+80x−67. It is given that there exist real numbers A, B, and C such that s3−22s2+80s−671=s−pA+s−qB+s−rC for all s∈{p,q,r}. What is A1+B1+C1?<spanclass=′latex−bold′>(A)</span>243<spanclass=′latex−bold′>(B)</span>244<spanclass=′latex−bold′>(C)</span>245<spanclass=′latex−bold′>(D)</span>246<spanclass=′latex−bold′>(E)</span>247 Real Analysis on AMC
Define a sequence recursively by x0=5 and
xn+1=xn+6xn2+5xn+4
for all nonnegative integers n. Let m be the least positive integer such that
xm≤4+2201. In which of the following intervals does m lie?<spanclass=′latex−bold′>(A)</span>[9,26]<spanclass=′latex−bold′>(B)</span>[27,80]<spanclass=′latex−bold′>(C)</span>[81,242]<spanclass=′latex−bold′>(D)</span>[243,728]<spanclass=′latex−bold′>(E)</span>[729,∞] Factorial Madness
The base-ten representation for 19! is 121,6T5,100,40M,832,H00, where T, M, and H denote digits that are not given. What is T+M+H?<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>14<spanclass=′latex−bold′>(E)</span>17 Raashan, Sylvia, and Ted and Chip Firing
Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every 15 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 $1 to that player. What is the probability that after the bell has rung 2019 times, each player will have $1? (For example, Raashan and Ted may each decide to give $1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0, Sylvia would have $2, and Ted would have $1, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $1 to, and and the holdings will be the same as the end of the second [sic] round.<spanclass=′latex−bold′>(A)</span>71<spanclass=′latex−bold′>(B)</span>41<spanclass=′latex−bold′>(C)</span>31<spanclass=′latex−bold′>(D)</span>21<spanclass=′latex−bold′>(E)</span>32 Arithmetic Lines
All lines with equation ax+by=c such that a, b, c form an arithmetic progression pass through a common point. What are the coordinates of that point?<spanclass=′latex−bold′>(A)</span>(−1,2)<spanclass=′latex−bold′>(B)</span>(0,1)<spanclass=′latex−bold′>(C)</span>(1,−2)<spanclass=′latex−bold′>(D)</span>(1,0)<spanclass=′latex−bold′>(E)</span>(1,2)