2022 AMC and AIME proposals
by djmathman, Feb 18, 2022, 1:11 AM
Had quite a few this year. Probably will be the last year I have this many.
10A 2 / 12A 2 (Summer 2020). Menkara has a
index card. If she shortens the length of one side of this card by 
inch, the card would have area 
square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by inch?
12A 21 (April/May 2020). Let
be an isosceles trapezoid with 
and 
. Points 
and 
lie on diagonal 
with between 
and , as shown in the figure. Suppose 
, 
, 
, and 
. What is the area of 
![[asy] size(5cm); usepackage("mathptmx"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(0.8),q=black+linewidth(4.5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label("2",C--Y,S); label("1",Y--X,S); label("3",X--A,S); label("$A$",A,E); label("$B$",B,N); label("$C$",C,W); label("$D$",D,S); label("$Y$",Y,sqrt(2)*NE); label("$X$",X,N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]](//latex.artofproblemsolving.com/9/f/e/9fe428a0357350b7518ad394f7ec668980159957.png)
10B 4 (Summer 2020). At noon on a certain day, Minneapolis is
degrees warmer than St. Louis. At 
the temperature in Minneapolis has fallen by 
degrees while the temperature in St. Louis has risen by 
degrees, at which time the temperatures in the two cities differ by 
degrees. What is the product of all possible values of 
10B 15 (Summer 2019). In square, points 
and 
lie on 
and 
, respectively. Segments 
and 
intersect at right angles at 
, with 
and 
. What is the area of the square?
![[asy] size(130); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); dot("$Q$",Q,S); dot("$P$",P,W); dot("$R$",R,1.3*S); label("$7$",(P+R)/2,NE); label("$6$",(R+B)/2,NE); [/asy]](//latex.artofproblemsolving.com/c/d/b/cdbf61b5c0e0123252d0e8083f70f0065be70554.png)
12B 16 (March 2020). Let
and 
be positive integers such that 
and ![\[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\]](//latex.artofproblemsolving.com/c/8/6/c86314b09c49a0f977dc31b9885edaaceafe776a.png)
What is the sum of all possible distinct values of 
?
10B 25 (March 2020). A rectangle with side lengths
and 
a square with side length 
and a rectangle are inscribed inside a larger square as shown. The sum of all possible values for the area of can be written in the form 
, where 
and 
are relatively prime positive integers. What is 
![[asy] size(6cm); draw((0,0)--(10,0)); draw((0,0)--(0,10)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((1,6)--(0,9)); draw((0,9)--(3,10)); draw((3,10)--(4,7)); draw((4,7)--(1,6)); draw((0,3)--(1,6)); draw((1,6)--(10,3)); draw((10,3)--(9,0)); draw((9,0)--(0,3)); draw((6,13/3)--(10,22/3)); draw((10,22/3)--(8,10)); draw((8,10)--(4,7)); draw((4,7)--(6,13/3)); label("$3$",(9/2,3/2),N); label("$3$",(11/2,9/2),S); label("$1$",(1/2,9/2),E); label("$1$",(19/2,3/2),W); label("$1$",(1/2,15/2),E); label("$1$",(3/2,19/2),S); label("$1$",(5/2,13/2),N); label("$1$",(7/2,17/2),W); label("$R$",(7,43/6),W); [/asy]](//latex.artofproblemsolving.com/1/c/1/1c19996df638fe0bc8406530f63293be6a81ee9b.png)
AIME I 1 (December 2020). Quadratic polynomials
and 
have leading coefficients of and 
, respectively. The graphs of both polynomials pass through the two points 
and 
. Find 
.
AIME I 11 (May/June 2020). Let be a parallelogram with 
. A circle tangent to sides 
, , and 
intersects diagonal at points and with 
, as shown. Suppose that 
, 
, and 
. Then the area of can be expressed in the form 
, where and are positive integers, and is not divisible by the square of any prime. Find 
.
![[asy]
defaultpen(linewidth(0.6)+fontsize(11));
size(8cm);
pair A,B,C,D,P,Q;
A=(0,0);
label("$A$", A, SW);
B=(6,15);
label("$B$", B, NW);
C=(30,15);
label("$C$", C, NE);
D=(24,0);
label("$D$", D, SE);
P=(5.2,2.6);
label("$P$", (5.8,2.6), N);
Q=(18.3,9.1);
label("$Q$", (18.1,9.7), W);
draw(A--B--C--D--cycle);
draw(C--A);
draw(Circle((10.95,7.45), 7.45));
dot(A^^B^^C^^D^^P^^Q);
[/asy]](//latex.artofproblemsolving.com/9/4/7/9471215d85465568eba3e615c0538a62e755bcf8.png)
AIME II 5 (December 2019). Twenty distinct points are marked on a circle and labeled through 
in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original points.
AIME II 8 (April 2020). Find the number of positive integers
whose value can be uniquely determined when the values of 
, 
, and 
are given, where 
denotes the greatest integer less than or equal to the real number 
.
AIME II 11 (February 2020). Let be a convex quadrilateral with 
and 
such that the bisectors of acute angles 
and 
intersect at the midpoint of 
Find the square of the area of 
AIME II 15 (December 2019). Two externally tangent circles
and 
have centers 
and 
, respectively. A third circle 
passing through and intersects at 
and 
and at and 
, as shown. Suppose that 
, 
, 
, and 
is a convex hexagon. Find the area of this hexagon.
![[asy] import geometry; size(8cm); defaultpen(fontsize(11)); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]](//latex.artofproblemsolving.com/c/f/4/cf4366ae7bdff349332214d10a865c7d939d0960.png)
Overall, I'm quite happy with this group of problems
I personally believe my AIME proposals are stronger than my AMC proposals, but I'm pleasantly surprised at the reception to my AMC problems, too. (I thought 12B 16 was meh and feared people would hate me for 10B 25....) Personal favorites are probably 10B 15 and AIME II 15, but you could probably convince me that quite a few others are up there, too.
10A 2 / 12A 2 (Summer 2020). Menkara has a
index card. If she shortens the length of one side of this card by 
inch, the card would have area 
square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by inch?12A 21 (April/May 2020). Let
be an isosceles trapezoid with 
and 
. Points 
and 
lie on diagonal 
with between 
and , as shown in the figure. Suppose 
, 
, 
, and 
. What is the area of 
![[asy] size(5cm); usepackage("mathptmx"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(0.8),q=black+linewidth(4.5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label("2",C--Y,S); label("1",Y--X,S); label("3",X--A,S); label("$A$",A,E); label("$B$",B,N); label("$C$",C,W); label("$D$",D,S); label("$Y$",Y,sqrt(2)*NE); label("$X$",X,N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]](http://latex.artofproblemsolving.com/9/f/e/9fe428a0357350b7518ad394f7ec668980159957.png)
10B 4 (Summer 2020). At noon on a certain day, Minneapolis is
degrees warmer than St. Louis. At 
the temperature in Minneapolis has fallen by 
degrees while the temperature in St. Louis has risen by 
degrees, at which time the temperatures in the two cities differ by 
degrees. What is the product of all possible values of 
10B 15 (Summer 2019). In square
, points 
and 
lie on 
and 
, respectively. Segments 
and 
intersect at right angles at 
, with 
and 
. What is the area of the square?![[asy] size(130); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); dot("$Q$",Q,S); dot("$P$",P,W); dot("$R$",R,1.3*S); label("$7$",(P+R)/2,NE); label("$6$",(R+B)/2,NE); [/asy]](http://latex.artofproblemsolving.com/c/d/b/cdbf61b5c0e0123252d0e8083f70f0065be70554.png)
12B 16 (March 2020). Let
and 
be positive integers such that 
and ![\[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\]](http://latex.artofproblemsolving.com/c/8/6/c86314b09c49a0f977dc31b9885edaaceafe776a.png)
What is the sum of all possible distinct values of 
?10B 25 (March 2020). A rectangle with side lengths
and 
a square with side length 
and a rectangle are inscribed inside a larger square as shown. The sum of all possible values for the area of can be written in the form 
, where 
and 
are relatively prime positive integers. What is 
![[asy] size(6cm); draw((0,0)--(10,0)); draw((0,0)--(0,10)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((1,6)--(0,9)); draw((0,9)--(3,10)); draw((3,10)--(4,7)); draw((4,7)--(1,6)); draw((0,3)--(1,6)); draw((1,6)--(10,3)); draw((10,3)--(9,0)); draw((9,0)--(0,3)); draw((6,13/3)--(10,22/3)); draw((10,22/3)--(8,10)); draw((8,10)--(4,7)); draw((4,7)--(6,13/3)); label("$3$",(9/2,3/2),N); label("$3$",(11/2,9/2),S); label("$1$",(1/2,9/2),E); label("$1$",(19/2,3/2),W); label("$1$",(1/2,15/2),E); label("$1$",(3/2,19/2),S); label("$1$",(5/2,13/2),N); label("$1$",(7/2,17/2),W); label("$R$",(7,43/6),W); [/asy]](http://latex.artofproblemsolving.com/1/c/1/1c19996df638fe0bc8406530f63293be6a81ee9b.png)
AIME I 1 (December 2020). Quadratic polynomials
and 
have leading coefficients of and 
, respectively. The graphs of both polynomials pass through the two points 
and 
. Find 
.AIME I 11 (May/June 2020). Let
be a parallelogram with 
. A circle tangent to sides 
, , and 
intersects diagonal at points and with 
, as shown. Suppose that 
, 
, and 
. Then the area of can be expressed in the form 
, where and are positive integers, and is not divisible by the square of any prime. Find 
.![[asy]
defaultpen(linewidth(0.6)+fontsize(11));
size(8cm);
pair A,B,C,D,P,Q;
A=(0,0);
label("$A$", A, SW);
B=(6,15);
label("$B$", B, NW);
C=(30,15);
label("$C$", C, NE);
D=(24,0);
label("$D$", D, SE);
P=(5.2,2.6);
label("$P$", (5.8,2.6), N);
Q=(18.3,9.1);
label("$Q$", (18.1,9.7), W);
draw(A--B--C--D--cycle);
draw(C--A);
draw(Circle((10.95,7.45), 7.45));
dot(A^^B^^C^^D^^P^^Q);
[/asy]](http://latex.artofproblemsolving.com/9/4/7/9471215d85465568eba3e615c0538a62e755bcf8.png)
AIME II 5 (December 2019). Twenty distinct points are marked on a circle and labeled
through 
in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original points.AIME II 8 (April 2020). Find the number of positive integers
whose value can be uniquely determined when the values of 
, 
, and 
are given, where 
denotes the greatest integer less than or equal to the real number 
.AIME II 11 (February 2020). Let
be a convex quadrilateral with 
and 
such that the bisectors of acute angles 
and 
intersect at the midpoint of 
Find the square of the area of 
AIME II 15 (December 2019). Two externally tangent circles
and 
have centers 
and 
, respectively. A third circle 
passing through and intersects at 
and 
and at and 
, as shown. Suppose that 
, 
, 
, and 
is a convex hexagon. Find the area of this hexagon.![[asy] import geometry; size(8cm); defaultpen(fontsize(11)); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]](http://latex.artofproblemsolving.com/c/f/4/cf4366ae7bdff349332214d10a865c7d939d0960.png)
Overall, I'm quite happy with this group of problems
I personally believe my AIME proposals are stronger than my AMC proposals, but I'm pleasantly surprised at the reception to my AMC problems, too. (I thought 12B 16 was meh and feared people would hate me for 10B 25....) Personal favorites are probably 10B 15 and AIME II 15, but you could probably convince me that quite a few others are up there, too.This post has been edited 1 time. Last edited by djmathman, Feb 18, 2022, 1:13 AM
April 2025
November 2024