Subcontests
(25)sussy baka stop intersecting in my lattice points
Let R, S, and T be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of R and the right edge of S are on the y-axis, and R contains 49 as many lattice points as does S. The top two vertices of T are in R∪S, and T contains 41 of the lattice points contained in R∪S. See the figure (not drawn to scale).[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"y", (0,60), N);
label(scale(.8)*"x", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"R", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"S",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"T",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]The fraction of lattice points in S that are in S∩T is 27 times the fraction of lattice points in R that are in R∩T. What is the minimum possible value of the edge length of R plus the edge length of S plus the edge length of T?<spanclass=′latex−bold′>(A)</span>336<spanclass=′latex−bold′>(B)</span>337<spanclass=′latex−bold′>(C)</span>338<spanclass=′latex−bold′>(D)</span>339<spanclass=′latex−bold′>(E)</span>340 Index Cards :D
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be 42 centimeters, as shown below. What is the area of the original index card?[asy]
unitsize(0.6 cm);pair A, B, C, D, E, F, G, H;
real x, y;
x = 9;
y = 5;A = (0,y);
B = (x - 1,y);
C = (x - 1,y - 1);
D = (x,y - 1);
E = (x,0);
F = (1,0);
G = (1,1);
H = (0,1);draw(A--B--C--D--E--F--G--H--cycle);
draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6));
draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6));label("1", (B + C)/2, W);
label("1", (C + D)/2, S);
label("8", interp(A,E,0.3), NE);
label("42", interp(G,C,0.2), SE);
[/asy]<spanclass=′latex−bold′>(A)</span>14<spanclass=′latex−bold′>(B)</span>102<spanclass=′latex−bold′>(C)</span>16<spanclass=′latex−bold′>(D)</span>122<spanclass=′latex−bold′>(E)</span>18 bijection moment
How many strings of length 5 formed from the digits 0,1,2,3,4 are there such that for each j∈{1,2,3,4}, at least j of the digits are less than j? (For example, 02214 satisfies the condition because it contains at least 1 digit less than 1, at least 2 digits less than 2, at least 3 digits less than 3, and at least 4 digits less than 4. The string 23404 does not satisfy the condition because it does not contain at least 2 digits less than 2.)<spanclass=′latex−bold′>(A)</span>500<spanclass=′latex−bold′>(B)</span>625<spanclass=′latex−bold′>(C)</span>1089<spanclass=′latex−bold′>(D)</span>1199<spanclass=′latex−bold′>(E)</span>1296 2022 AMC 12B Problem 6
Consider the following 100 sets of 10 elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of 7?<spanclass=′latex−bold′>(A)</span>40<spanclass=′latex−bold′>(B)</span>42<spanclass=′latex−bold′>(C)</span>43<spanclass=′latex−bold′>(D)</span>49<spanclass=′latex−bold′>(E)</span>50 Painting a Rectangle
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]<spanclass=′latex−bold′>(A)</span>120<spanclass=′latex−bold′>(B)</span>270<spanclass=′latex−bold′>(C)</span>360<spanclass=′latex−bold′>(D)</span>540<spanclass=′latex−bold′>(E)</span>720 Card Games
Suppose that 13 cards numbered 1,2,3,…,13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the 13! possible orderings of the cards will the 13 cards be picked up in exactly two passes?[asy]
size(11cm);
draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
label("7", (1,1.5));
draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
label("11", (4,1.5));
draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("8", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("6", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
label("4", (13,1.5));
draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
label("5", (16,1.5));
draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
label("9", (19,1.5));
draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
label("12", (22,1.5));
draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
label("1", (25,1.5));
draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
label("13", (28,1.5));
draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
label("10", (31,1.5));
draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
label("2", (34,1.5));
draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
label("3", (37,1.5));
[/asy]<spanclass=′latex−bold′>(A)</span>4082<spanclass=′latex−bold′>(B)</span>4095<spanclass=′latex−bold′>(C)</span>4096<spanclass=′latex−bold′>(D)</span>8178<spanclass=′latex−bold′>(E)</span>8191 2022 AMC 10 A Problem 15
Quadrilateral ABCD with side lengths AB=7,BC=24,CD=20,DA=15 is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form caπ−b, where a,b, and c are positive integers such that a and c have no common prime factor. What is a+b+c?<spanclass=′latex−bold′>(A)</span>260<spanclass=′latex−bold′>(B)</span>855<spanclass=′latex−bold′>(C)</span>1235<spanclass=′latex−bold′>(D)</span>1565<spanclass=′latex−bold′>(E)</span>1997 Rectangle and Square
The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("4", (8,2), E);
label("8", (4,0), S);
label("5", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]<spanclass=′latex−bold′>(A)</span>1581<spanclass=′latex−bold′>(B)</span>1583<spanclass=′latex−bold′>(C)</span>1521<spanclass=′latex−bold′>(D)</span>1585<spanclass=′latex−bold′>(E)</span>1587 Diagrams and 3D Geo
A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?[asy]
size(200);
defaultpen(linewidth(0.8));
draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300));
draw((82,-204) -- (133,-490) -- (342, -662));
draw((652,-626) -- (600,-727));
draw((447,-458) -- (652,-626) -- (859,-410));
draw((133,-490) -- (184, -388));
draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4));
[/asy]<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>7<spanclass=′latex−bold′>(C)</span>5+22<spanclass=′latex−bold′>(D)</span>8<spanclass=′latex−bold′>(E)</span>9 Subset Logic
Suppose that S is a subset of {1,2,3,...,25} such that the sum of any two (not necessarily distinct) elements of S is never an element of S. What is the maximum number of elements S may contain?<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>13<spanclass=′latex−bold′>(C)</span>14<spanclass=′latex−bold′>(D)</span>15<spanclass=′latex−bold′>(E)</span>16 MAA is cruel
Each square in a 5×5 grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:[*] Any filled square with two or three filled neighbors remains filled.
[*] Any empty square with exactly three filled neighbors becomes a filled square.
[*] All other squares remain empty or become empty.A sample transformation is shown in the figure below.[asy]
import geometry;
unitsize(0.6cm); void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
} ds((1,1));
ds((2,1));
ds((3,1));
ds((1,3)); for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
} label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow); ds((10,2));
ds((11,1));
ds((11,0)); for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
} label("Transformed", (11.5,-1));
[/asy]Suppose the 5×5 grid has a border of empty squares surrounding a 3×3 subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
[asy]
import geometry;
unitsize(0.6cm); void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
} for (int i = 1; i < 4; ++ i) {
for (int j = 1; j < 4; ++j) {
label("?",(i + 0.5, j + 0.5));
}
} for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
} label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow); ds((11,2)); for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
} label("Transformed", (11.5,-1));
[/asy]<spanclass=′latex−bold′>(A)14</span> <spanclass=′latex−bold′>(B)18</span> <spanclass=′latex−bold′>(C)22</span> <spanclass=′latex−bold′>(D)26</span> <spanclass=′latex−bold′>(E)30</span> Simple Rhombus
In rhombus ABCD, point P lies on segment AD such that BP⊥AD, AP=3, and PD=2. What is the area of ABCD? [asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("A",A,SW);
label("B", B, NW);
label("C",C,NE);
label("D",D,SE);
label("P",P,S);
[/asy]
<spanclass=′latex−bold′>(A)</span>35<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>65<spanclass=′latex−bold′>(D)</span>20<spanclass=′latex−bold′>(E)</span>25 Coordinate Plane
Let Tk be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the origin and then reflects the plane across the y-axis. What is the least positive integer n such that performing the sequence of transformations transformations T1,T2,T3,…,Tn returns the point (1,0) back to itself?<spanclass=′latex−bold′>(A)</span>359<spanclass=′latex−bold′>(B)</span>360<spanclass=′latex−bold′>(C)</span>719<spanclass=′latex−bold′>(D)</span>720<spanclass=′latex−bold′>(E)</span>721 linear algebra time i guess
Consider systems of three linear equations with unknowns x, y, and z,
\begin{align*}
a_1 x + b_1 y + c_1 z = 0 \\
a_2 x + b_2 y + c_2 z = 0 \\
a_3 x + b_3 y + c_3 z = 0
\end{align*}
where each of the coefficients is either 0 or 1 and the system has a solution other than x=y=z=0. For example, one such system is {1x+1y+0z=0,0x+1y+1z=0,0x+0y+0z=0} with a nonzero solution of {x,y,z}={1,−1,1}. How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)<spanclass=′latex−bold′>(A)</span>302<spanclass=′latex−bold′>(B)</span>338<spanclass=′latex−bold′>(C)</span>340<spanclass=′latex−bold′>(D)</span>343<spanclass=′latex−bold′>(E)</span>344 Isosceles
Isosceles trapezoid ABCD has parallel sides AD and BC, with BC<AD and AB=CD. There is a point P in the plane such that PA=1,PB=2,PC=3, and PD=4. What is ADBC?<spanclass=′latex−bold′>(A)</span>41<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>21<spanclass=′latex−bold′>(D)</span>32<spanclass=′latex−bold′>(E)</span>43 AMELIA ATEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Ant Amelia starts on the number line at 0 and crawls in the following manner. For n=1,2,3, Amelia chooses a time duration tn and an increment xn independently and uniformly at random from the interval (0,1). During the nth step of the process, Amelia moves xn units in the positive direction, using up tn minutes. If the total elapsed time has exceeded 1 minute during the nth step, she stops at the end of that step; otherwise, she continues with the next step, taking at most 3 steps in all. What is the probability that Amelia’s position when she stops will be greater than 1?<spanclass=′latex−bold′>(A)</span>31<spanclass=′latex−bold′>(B)</span>21<spanclass=′latex−bold′>(C)</span>32<spanclass=′latex−bold′>(D)</span>43<spanclass=′latex−bold′>(E)</span>65