2024 AMC Proposals
by djmathman, Nov 13, 2024, 11:17 PM
Four proposals this time. One great, three OK.
12A 12/10A 19. The first three terms of a geometric sequence are the integers
and 
where 
What is the sum of the digits of the least possible value of 
10A 22. Let
be the kite formed by joining two right triangles with legs 
and 
along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle 
?
10A 25/12A 22. The figure below shows a dotted grid
cells wide and 
cells tall consisting of 
squares. Carl places -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? ![[asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy]](//latex.artofproblemsolving.com/b/1/c/b1cfd38f58057d8f0ed435b054143237ba4fac8c.png)
12B 19. Equilateral with side length 
is rotated about its center by angle 
, where 
, to form 
. See the figure. The area of hexagon 
is 
. What is 
?
![[asy]
defaultpen(fontsize(13)); size(170);
pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C;
draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F));
[/asy]](//latex.artofproblemsolving.com/a/7/2/a72db31d1d76e376654dd13a3f9a166d0d6056ce.png)
12A 12/10A 19. The first three terms of a geometric sequence are the integers
and 
where 
What is the sum of the digits of the least possible value of 
is a good choice, because once you stumble upon 
it's pretty easy to convince yourself this is optimal. But this problem doesn't really have the same "impact" as some of my previous proposals.10A 22. Let
be the kite formed by joining two right triangles with legs 
and 
along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle 
?
,
,
was intentional: I wanted to make sure most (if not all) of the kites were part of the triangle.10A 25/12A 22. The figure below shows a dotted grid
cells wide and 
cells tall consisting of 
squares. Carl places -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? ![[asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy]](http://latex.artofproblemsolving.com/b/1/c/b1cfd38f58057d8f0ed435b054143237ba4fac8c.png)
s12B 19. Equilateral
with side length 
is rotated about its center by angle 
, where 
, to form 
. See the figure. The area of hexagon 
is 
. What is 
?![[asy]
defaultpen(fontsize(13)); size(170);
pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C;
draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F));
[/asy]](http://latex.artofproblemsolving.com/a/7/2/a72db31d1d76e376654dd13a3f9a166d0d6056ce.png)
This post has been edited 2 times. Last edited by djmathman, Nov 13, 2024, 11:19 PM
April 2025
November 2024