Subcontests
(20)AMC 12B #3
Which one of the following rigid transformations (isometries) maps the line segment AB onto the line segment A′B′ so that the image of A(−2,1) is A′(2,−1) and the image of B(−1,4) is B′(1,−4)? <spanclass=′latex−bold′>(A)</span> reflection in the y-axis
<spanclass=′latex−bold′>(B)</span> counterclockwise rotation around the origin by 90∘
<spanclass=′latex−bold′>(C)</span> translation by 3 units to the right and 5 units down
<spanclass=′latex−bold′>(D)</span> reflection in the x-axis
<spanclass=′latex−bold′>(E)</span> clockwise rotation about the origin by 180∘ Painful Paths
The figure below is a map showing 12 cities and 17 roads connecting certain pairs of cities. Paula wishes to travel along exactly 13 of those roads, starting at city A and ending at city L, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) How many different routes can Paula take?[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]<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 3d projective geo?
Square pyramid ABCDE has base ABCD, which measures 3 cm on a side, and altitude AE perpendicular to the base, which measures 6 cm. Point P lies on BE, one third of the way from B to E; point Q lies on DE, one third of the way from D to E; and point R lies on CE, two thirds of the way from C to E. What is the area, in square centimeters, of △PQR?<spanclass=′latex−bold′>(A)</span>232<spanclass=′latex−bold′>(B)</span>233<spanclass=′latex−bold′>(C)</span>22<spanclass=′latex−bold′>(D)</span>23<spanclass=′latex−bold′>(E)</span>32 Right triangles (2019 AMC 12B #12)
Right triangle ACD with right angle at C is constructed outwards on the hypotenuse AC of isosceles right triangle ABC with leg length 1, as shown, so that the two triangles have equal perimeters. What is sin(2∠BAD)?
[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]
<spanclass=′latex−bold′>(A)</span>31<spanclass=′latex−bold′>(B)</span>22<spanclass=′latex−bold′>(C)</span>43<spanclass=′latex−bold′>(D)</span>97<spanclass=′latex−bold′>(E)</span>23 Equilateral triangle and circles
Circles ω and γ, both centered at O, have radii 20 and 17, respectively. Equilateral triangle ABC, whose interior lies in the interior of ω but in the exterior of γ, has vertex A on ω, and the line containing side BC is tangent to γ. Segments AO and BC intersect at P, and CPBP=3. Then AB can be written in the form nm−qp for positive integers m, n, p, q with gcd(m,n)=gcd(p,q)=1. What is m+n+p+q?
<spanclass=′latex−bold′>(A)</span>42<spanclass=′latex−bold′>(B)</span>86<spanclass=′latex−bold′>(C)</span>92<spanclass=′latex−bold′>(D)</span>114<spanclass=′latex−bold′>(E)</span>130 Roots equals Coefficients
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 ax2+bx+c,a=0, and the roots are r and s, then the requirement is that {a,b,c}={r,s}.)<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>infinitely many Sums Powers of Roots
Let sk denote the sum of the <spanclass=′latex−italic′>k</span>th powers of the roots of the polynomial x3−5x2+8x−13. In particular, s0=3, s1=5, and s2=9. Let a, b, and c be real numbers such that sk+1=ask+bsk−1+csk−2 for k=2, 3, .... What is a+b+c?<spanclass=′latex−bold′>(A)</span>−6<spanclass=′latex−bold′>(B)</span>0<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>26 Rotating in the complex plane
How many nonzero complex numbers z have the property that 0,z, and z3, when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>4<spanclass=′latex−bold′>(E)</span>infinitely many Binary Operations and Logarithms
Define binary operations ♢ and ♡ by a♢b=alog7(b)anda♡b=alog7(b)1
for all real numbers a and b for which these expressions are defined. The sequence (an) is defined recursively by a3=3♡2 and an=(n♡(n−1))♢an−1
for all integers n≥4. To the nearest integer, what is log7(a2019)?<spanclass=′latex−bold′>(A)</span>8<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>10<spanclass=′latex−bold′>(D)</span>11<spanclass=′latex−bold′>(E)</span>12 Painting the Integers With Respect To Its Proper Divisors
How many ways are there to paint each of the integers 2,3,…,9 either red, green, or blue so that each number has a different color from each of its proper divisors?<spanclass=′latex−bold′>(A)</span> 144<spanclass=′latex−bold′>(B)</span> 216<spanclass=′latex−bold′>(C)</span> 256<spanclass=′latex−bold′>(D)</span> 384<spanclass=′latex−bold′>(E)</span> 432