2021 AMC B Proposals
by djmathman, Feb 11, 2021, 7:35 PM
altigeo strikes again :O
10B 5 (Summer 2018 - Summer 2019). The ages of Jonie’s four cousins are distinct single-digit positive integers. Two of the cousins’ ages multiplied together give
, while the other two multiply to 
. What is the sum of the ages of Jonie’s four cousins?
10B 20/12B 15 (Spring 2019). The figure below is constructed from
line segments, each of which has length 
. The area of pentagon 
can be written as 
, where 
and 
are positive integers. What is 
?
![[asy]
pair A=(-2.4638,4.10658);
pair B=(-4,2.6567453480756127);
pair C=(-3.47132,0.6335248637894945);
pair D=(-1.464483379039766,0.6335248637894945);
pair E=(-0.956630463955801,2.6567453480756127);
pair F=(-2,2);
pair G=(-3,2);
draw(A--B--C--D--E--A);
draw(A--F--A--G);
draw(B--F--C);
draw(E--G--D);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,SW);
label("$D$",D,SE);
label("$E$",E,dir(0));
dot(A^^B^^C^^D^^E^^F^^G);
[/asy]](//latex.artofproblemsolving.com/6/6/a/66a5b175cc51c8751f8a524f71a5c2f039868ec2.png)
12B 17 (Spring 2019). Let
be an isoceles trapezoid having parallel bases 
and 
with 
Line segments from a point inside to the vertices divide the trapezoid into four triangles whose areas are 
and 
starting with the triangle with base and moving clockwise as shown in the diagram below. What is the ratio 
![[asy]unitsize(100);
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5);
draw(A--B--C--D--cycle, black);
draw(A--P, black);
draw(B--P, black);
draw(C--P, black);
draw(D--P, black);
label("$A$",A,(-1,0));
label("$B$",B,(1,0));
label("$C$",C,(1,-0));
label("$D$",D,(-1,0));
label("$2$",E,(0,0));
label("$3$",F,(0,0));
label("$4$",G,(0,0));
label("$5$",H,(0,0));
dot(A^^B^^C^^D^^P);
[/asy]](//latex.artofproblemsolving.com/8/d/a/8da767b461d070528eb6a05d45d906dc1d08a899.png)
12B 24 (Summer 2018). Let be a parallelogram with area 
. Points 
and 
are the projections of 
and 
respectively, onto the line 
and points 
and 
are the projections of 
and 
respectively, onto the line 
See the figure, which also shows the relative locations of these points.
![[asy]
size(350);
defaultpen(linewidth(0.8)+fontsize(11));
real theta = aTan(1.25/2);
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));
dot("$A$",A,dir(270));
dot("$B$",B,E);
dot("$C$",C,N);
dot("$D$",D,W);
dot("$P$",P,SE);
dot("$Q$",Q,NE);
dot("$R$",R,N);
dot("$S$",S,dir(270));
[/asy]](//latex.artofproblemsolving.com/1/4/6/146f1376f1f3d196ea8f9f85dbc4decfcac840b9.png)
Suppose
and 
and let 
denote the length of 
the longer diagonal of 
Then 
can be written in the form 
where 
and 
are positive integers and is not divisible by the square of any prime. What is 
. This batch was probably my strongest yet.
10B 5 (Summer 2018 - Summer 2019). The ages of Jonie’s four cousins are distinct single-digit positive integers. Two of the cousins’ ages multiplied together give
, while the other two multiply to 
. What is the sum of the ages of Jonie’s four cousins?10B 20/12B 15 (Spring 2019). The figure below is constructed from
line segments, each of which has length 
. The area of pentagon 
can be written as 
, where 
and 
are positive integers. What is 
?![[asy]
pair A=(-2.4638,4.10658);
pair B=(-4,2.6567453480756127);
pair C=(-3.47132,0.6335248637894945);
pair D=(-1.464483379039766,0.6335248637894945);
pair E=(-0.956630463955801,2.6567453480756127);
pair F=(-2,2);
pair G=(-3,2);
draw(A--B--C--D--E--A);
draw(A--F--A--G);
draw(B--F--C);
draw(E--G--D);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,SW);
label("$D$",D,SE);
label("$E$",E,dir(0));
dot(A^^B^^C^^D^^E^^F^^G);
[/asy]](http://latex.artofproblemsolving.com/6/6/a/66a5b175cc51c8751f8a524f71a5c2f039868ec2.png)
12B 17 (Spring 2019). Let
be an isoceles trapezoid having parallel bases 
and 
with 
Line segments from a point inside to the vertices divide the trapezoid into four triangles whose areas are 
and 
starting with the triangle with base and moving clockwise as shown in the diagram below. What is the ratio 
![[asy]unitsize(100);
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5);
draw(A--B--C--D--cycle, black);
draw(A--P, black);
draw(B--P, black);
draw(C--P, black);
draw(D--P, black);
label("$A$",A,(-1,0));
label("$B$",B,(1,0));
label("$C$",C,(1,-0));
label("$D$",D,(-1,0));
label("$2$",E,(0,0));
label("$3$",F,(0,0));
label("$4$",G,(0,0));
label("$5$",H,(0,0));
dot(A^^B^^C^^D^^P);
[/asy]](http://latex.artofproblemsolving.com/8/d/a/8da767b461d070528eb6a05d45d906dc1d08a899.png)
12B 24 (Summer 2018). Let
be a parallelogram with area 
. Points 
and 
are the projections of 
and 
respectively, onto the line 
and points 
and 
are the projections of 
and 
respectively, onto the line 
See the figure, which also shows the relative locations of these points.![[asy]
size(350);
defaultpen(linewidth(0.8)+fontsize(11));
real theta = aTan(1.25/2);
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));
dot("$A$",A,dir(270));
dot("$B$",B,E);
dot("$C$",C,N);
dot("$D$",D,W);
dot("$P$",P,SE);
dot("$Q$",Q,NE);
dot("$R$",R,N);
dot("$S$",S,dir(270));
[/asy]](http://latex.artofproblemsolving.com/1/4/6/146f1376f1f3d196ea8f9f85dbc4decfcac840b9.png)
Suppose
and 
and let 
denote the length of 
the longer diagonal of 
Then 
can be written in the form 
where 
and 
are positive integers and is not divisible by the square of any prime. What is 
. This batch was probably my strongest yet.This post has been edited 5 times. Last edited by djmathman, Feb 11, 2021, 11:35 PM
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