Subcontests
(25)New Coord Bash Problem
Rectangle ABCD has AB=5 and BC=4. Point E lies on AB so that EB=1, point G lies on BC so that CG=1. and point F lies on CD so that DF=2. Segments AG and AC intersect EF at Q and P, respectively. What is the value of EFPQ?
[asy] pair A1=(2,0),A2=(4,4);
pair B1=(0,4),B2=(5,1);
pair C1=(5,0),C2=(0,4);
draw(A1--A2);
draw(B1--B2);
draw(C1--C2);
draw((0,0)--B1--(5,4)--C1--cycle);
dot((20/7,12/7));
dot((3.07692307692,2.15384615384));
label("Q",(3.07692307692,2.15384615384),N);
label("P",(20/7,12/7),W);
label("A",(0,4), NW);
label("B",(5,4), NE);
label("C",(5,0),SE);
label("D",(0,0),SW);
label("F",(2,0),S); label("G",(5,1),E);
label("E",(4,4),N);dot(A1); dot(A2);
dot(B1); dot(B2);
dot(C1); dot(C2);
dot((0,0)); dot((5,4));[/asy]<spanclass=′latex−bold′>(A)</span> 1613<spanclass=′latex−bold′>(B)</span> 132<spanclass=′latex−bold′>(C)</span> 829<spanclass=′latex−bold′>(D)</span> 9110<spanclass=′latex−bold′>(E)</span> 91 Triangle
A triangle with vertices A(0,2), B(−3,2), and C(−3,0) is reflected about the x-axis, then the image △A′B′C′ is rotated counterclockwise about the origin by 90∘ to produce △A′′B′′C′′. Which of the following transformations will return △A′′B′′C′′ to △ABC?<spanclass=′latex−bold′>(A)</span> counterclockwise rotation about the origin by 90∘.
<spanclass=′latex−bold′>(B)</span> clockwise rotation about the origin by 90∘.
<spanclass=′latex−bold′>(C)</span> reflection about the x-axis
<spanclass=′latex−bold′>(D)</span> reflection about the line y=x
<spanclass=′latex−bold′>(E)</span> reflection about the y-axis. Tri-colored rug
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
path rectangle(pair X, pair Y){
return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle;
}
filldraw(rectangle((0,0),(7,5)),gray(0.5));
filldraw(rectangle((1,1),(6,4)),gray(0.75));
filldraw(rectangle((2,2),(5,3)),white);label("1",(0.5,2.5));
draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead));
draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead));label("1",(1.5,2.5));
draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead));
draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead));label("1",(4.5,2.5));
draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead));
draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead));label("1",(4.1,1.5));
draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead));
draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead));label("1",(3.7,0.5));
draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead));
draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead));
[/asy]<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>2<spanclass=′latex−bold′>(C)</span>4<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>8 Weird Area
What is the area of the shaded region of the given 8×5 rectangle?[asy]size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));label("1",(1/2,5),dir(90));
label("7",(9/2,5),dir(90));label("1",(8,1/2),dir(0));
label("4",(8,3),dir(0));label("1",(15/2,0),dir(270));
label("7",(7/2,0),dir(270));label("1",(0,9/2),dir(180));
label("4",(0,2),dir(180));[/asy]<spanclass=′latex−bold′>(A)</span> 453<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 541<spanclass=′latex−bold′>(D)</span> 621<spanclass=′latex−bold′>(E)</span> 8 Circles in circles
Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?[asy]
draw(circle((0,0),3));
draw(circle((0,0),1));
draw(circle((1,sqrt(3)),1));
draw(circle((-1,sqrt(3)),1));
draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1));
draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy]<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>1.5<spanclass=′latex−bold′>(C)</span>π<spanclass=′latex−bold′>(D)</span>2π<spanclass=′latex−bold′>(E)</span>π 3 by 3 Array Of Squares
All the numbers 1,2,3,4,5,6,7,8,9 are written in a 3×3 array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18. What is the number in the center?<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 9 Equivalent averages
The mean, median, and mode of the 7 data values 60,100,x,40,50,200,90 are all equal to x. What is the value of x?
<spanclass=′latex−bold′>(A)</span> 50<spanclass=′latex−bold′>(B)</span> 60<spanclass=′latex−bold′>(C)</span> 75<spanclass=′latex−bold′>(D)</span> 90<spanclass=′latex−bold′>(E)</span> 100 Even Products
Two different numbers are selected at random from (1,2,3,4,5) and multiplied together. What is the probability that the product is even?<spanclass=′latex−bold′>(A)</span> 0.2<spanclass=′latex−bold′>(B)</span> 0.4<spanclass=′latex−bold′>(C)</span> 0.5<spanclass=′latex−bold′>(D)</span> 0.7<spanclass=′latex−bold′>(E)</span> 0.8
AMC 10B 2016 #17
All the numbers 2,3,4,5,6,7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?<spanclass=′latex−bold′>(A)</span> 312<spanclass=′latex−bold′>(B)</span> 343<spanclass=′latex−bold′>(C)</span> 625<spanclass=′latex−bold′>(D)</span> 729<spanclass=′latex−bold′>(E)</span> 1680 Awesome Binary Operation
A binary operation ♢ has the properties that a♢(b♢c)=(a♢b)⋅c and that a♢a=1 for all nonzero real numbers a,b, and c. (Here ⋅ represents multiplication). The solution to the equation 2016♢(6♢x)=100 can be written as qp, where p and q are relatively prime positive integers. What is p+q?<spanclass=′latex−bold′>(A)</span>109<spanclass=′latex−bold′>(B)</span>201<spanclass=′latex−bold′>(C)</span>301<spanclass=′latex−bold′>(D)</span>3049<spanclass=′latex−bold′>(E)</span>33,601 Hexagon Ratios
In regular hexagon ABCDEF, points W, X, Y, and Z are chosen on sides BC, CD, EF, and FA respectively, so lines AB, ZW, YX, and ED are parallel and equally spaced. What is the ratio of the area of hexagon WCXYFZ to the area of hexagon ABCDEF?<spanclass=′latex−bold′>(A)</span> 31<spanclass=′latex−bold′>(B)</span> 2710<spanclass=′latex−bold′>(C)</span> 2711<spanclass=′latex−bold′>(D)</span> 94<spanclass=′latex−bold′>(E)</span> 2713
Three LCM equations
How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y)=72,lcm(x,z)=600 and lcm(y,z)=900?<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 27<spanclass=′latex−bold′>(E)</span> 64 Three Externally Tangent Circles
Circles with centers P,Q and R, having radii 1,2 and 3, respectively, lie on the same side of line l and are tangent to l at P′,Q′ and R′, respectively, with Q′ between P′ and R′. The circle with center Q is externally tangent to each of the other two circles. What is the area of triangle PQR?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>32<spanclass=′latex−bold′>(C)</span>1<spanclass=′latex−bold′>(D)</span>6−2<spanclass=′latex−bold′>(E)</span>23 Crazy powers
For some particular value of N, when (a+b+c+d+1)N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a,b,c, and d, each to some positive power. What is N?<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>14<spanclass=′latex−bold′>(C)</span>16<spanclass=′latex−bold′>(D)</span>17<spanclass=′latex−bold′>(E)</span>19 Dilation Of The Plane
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius 2 centered at A(2,2) to the circle of radius 3 centered at A’(5,6). What distance does the origin O(0,0), move under this transformation?<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 13<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5