AMC 10 2020
A
2
The numbers 
and 
have an average (arithmetic mean) of 
. What is the average of 
and ?
and 
have an average (arithmetic mean) of 
. What is the average of 
and ?
3
Assuming 
, 
, and 
, what is the value in simplest form of the following expression?
, 
, and 
, what is the value in simplest form of the following expression?
4
A driver travels for 
hours at 
miles per hour, during which her car gets 
miles per gallon of gasoline. She is paid 
per mile, and her only expense is gasoline at 
per gallon. What is her net rate of pay, in dollars per hour, after this expense?
hours at 
miles per hour, during which her car gets 
miles per gallon of gasoline. She is paid 
per mile, and her only expense is gasoline at 
per gallon. What is her net rate of pay, in dollars per hour, after this expense?
6
How many 
-digit positive integers (that is, integers between 
and 
, inclusive) having only even digits are divisible by 
-digit positive integers (that is, integers between 
and 
, inclusive) having only even digits are divisible by 
7
The 
integers from 
to 
inclusive, can be arranged to form a 
-by- square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
integers from 
to 
inclusive, can be arranged to form a 
-by- square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
![\[1+2+3-4+5+6+7-8+\cdots+197+198+199-200?\]](http://latex.artofproblemsolving.com/6/5/4/6546d929f5a85740556d362a0491e8e13a4b9948.png)
9
A single bench section at a school event can hold either 
adults or 
children. When 
bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of 
adults or 
children. When 
bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of 
10
Seven cubes, whose volumes are 
, 
, 
, 
, 
, 
, and 
cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
, 
, 
, 
, 
, 
, and 
cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
number
12
Triangle 
is isoceles with 
. Medians 
and 
are perpendicular to each other, and 
. What is the area of 
![[asy]
draw((-4,0)--(4,0)--(0,12)--cycle);
draw((-2,6)--(4,0));
draw((2,6)--(-4,0));
draw((-2,6)--(2,6));
label("M", (-4,0), W);
label("C", (4,0), E);
label("A", (0, 12), N);
label("V", (2, 6), NE);
label("U", (-2, 6), NW);
draw(rightanglemark((-2,6),(0,4),(-4,0),17));
[/asy]](//latex.artofproblemsolving.com/c/1/2/c12a3d965ab244d0ac95bdc9974aecc8c0df3608.png)
is isoceles with 
. Medians 
and 
are perpendicular to each other, and 
. What is the area of 
![[asy]
draw((-4,0)--(4,0)--(0,12)--cycle);
draw((-2,6)--(4,0));
draw((2,6)--(-4,0));
draw((-2,6)--(2,6));
label("M", (-4,0), W);
label("C", (4,0), E);
label("A", (0, 12), N);
label("V", (2, 6), NE);
label("U", (-2, 6), NW);
draw(rightanglemark((-2,6),(0,4),(-4,0),17));
[/asy]](http://latex.artofproblemsolving.com/c/1/2/c12a3d965ab244d0ac95bdc9974aecc8c0df3608.png)
13
A frog sitting at the point 
begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices 
and 
. What is the probability that the sequence of jumps ends on a vertical side of the square
begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices 
and 
. What is the probability that the sequence of jumps ends on a vertical side of the square
and 
.![\[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\]](http://latex.artofproblemsolving.com/1/b/7/1b77f92edbab9a6134e14217ee0c56700cef3dbf.png)
15
A positive integer divisor of 
is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as 
, where 
and 
are relatively prime positive integers. What is 
?
is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as 
, where 
and 
are relatively prime positive integers. What is 
?
16
A point is chosen at random within the square in the coordinate plane whose vertices are 
and 
. The probability that the point is within 
units of a lattice point is 
. (A point 
is a lattice point if 
and 
are both integers.) What is to the nearest tenth
and 
. The probability that the point is within 
units of a lattice point is 
. (A point 
is a lattice point if 
and 
are both integers.) What is to the nearest tenth
How many integers 
?
18
Let 
be an ordered quadruple of not necessarily distinct integers, each one of them in the set 
. For how many such quadruples is it true that 
is odd (For example, 
is one such quadruple, because 
is odd.)
be an ordered quadruple of not necessarily distinct integers, each one of them in the set 
. For how many such quadruples is it true that 
is odd (For example, 
is one such quadruple, because 
is odd.)
19
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
![[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]](//latex.artofproblemsolving.com/a/1/f/a1f90b26e39b21c8049d1f5f6e8acc8887dfe339.png)
![[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]](http://latex.artofproblemsolving.com/a/1/f/a1f90b26e39b21c8049d1f5f6e8acc8887dfe339.png)
20
Quadrilateral 
satisfies 
, and 
. Diagonals 
and 
intersect at point 
, and 
. What is the area of quadrilateral ?
satisfies 
, and 
. Diagonals 
and 
intersect at point 
, and 
. What is the area of quadrilateral ?
21
There exists a unique strictly increasing sequence of nonnegative integers 
such that ![\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\]](//latex.artofproblemsolving.com/5/c/d/5cdaf7cc54fe0f813d490f315e6982697b67ad9f.png)
What is 
such that ![\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\]](http://latex.artofproblemsolving.com/5/c/d/5cdaf7cc54fe0f813d490f315e6982697b67ad9f.png)
What is 
22
For how many positive integers 
is 
not divisible by 
? (Recall that 
is the greatest integer less than or equal to .)
is 
not divisible by 
? (Recall that 
is the greatest integer less than or equal to .)
23
Let 
be the triangle in the coordinate plane with vertices 
, 
, and 
. Consider the following five isometries (rigid transformations) of the plane: rotations of 
, 
, and 
counterclockwise around the origin, reflection across the -axis, and reflection across the -axis. How many of the sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection across the -axis, followed by another reflection across the -axis will not return to its original position.)
be the triangle in the coordinate plane with vertices 
, 
, and 
. Consider the following five isometries (rigid transformations) of the plane: rotations of 
, 
, and 
counterclockwise around the origin, reflection across the -axis, and reflection across the -axis. How many of the sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection across the -axis, followed by another reflection across the -axis will not return to its original position.)
24
Let be the least positive integer greater than for which 
What is the sum of the digits of ?
be the least positive integer greater than for which 
What is the sum of the digits of ?
25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
B
2
Carl has cubes each having side length , and Kate has cubes each having side length . What is the total volume of the 
cubes?
cubes each having side length , and Kate has cubes each having side length . What is the total volume of the 
cubes?
3
The ratio of 
to is 
, the ratio of to 
is 
, and the ratio of to is 
. What is the ratio of to ?
to is 
, the ratio of to 
is 
, and the ratio of to is 
. What is the ratio of to ?
4
The acute angles of a right triangle are 
and 
, where 
and both and are prime numbers. What is the least possible value of ?
and 
, where 
and both and are prime numbers. What is the least possible value of ?
5
How many distinguishable arrangements are there of brown tile, purple tile, green tiles, and yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable)
brown tile, purple tile, green tiles, and yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable)
6
Driving along a highway, Megan noticed that her odometer showed 
(miles). This number is a palindrome—it reads the same forward and backward. Then hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this -hour period?
(miles). This number is a palindrome—it reads the same forward and backward. Then hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this -hour period?
8
Points 
and 
lie in a plane with 
. How many locations for point 
in this plane are there such that the triangle with vertices 
and is a right triangle with area 
square units?
and 
lie in a plane with 
. How many locations for point 
in this plane are there such that the triangle with vertices 
and is a right triangle with area 
square units?
10
A three-quarter sector of a circle of radius inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
![[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]](//latex.artofproblemsolving.com/3/2/6/32649d4753b5a48a0eba47330652ed5fc5a401a7.png)
inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?![[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]](http://latex.artofproblemsolving.com/3/2/6/32649d4753b5a48a0eba47330652ed5fc5a401a7.png)
11
Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?
12
The decimal representation of 
consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
13
Andy the Ant lives on a coordinate plane and is currently at 
facing east (that is, in the positive -direction). Andy moves unit and then turns degrees left. From there, Andy moves units (north) and then turns degrees left. He then moves units (west) and again turns degrees left. Andy continues his progress, increasing his distance each time by unit and always turning left. What is the location of the point at which Andy makes the 
th left turn?
facing east (that is, in the positive -direction). Andy moves unit and then turns degrees left. From there, Andy moves units (north) and then turns degrees left. He then moves units (west) and again turns degrees left. Andy continues his progress, increasing his distance each time by unit and always turning left. What is the location of the point at which Andy makes the 
th left turn?
14
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?
![[asy]
size(140);
fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4));
fill(arc((2,0),1,180,0)--(2,0)--cycle,white);
fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white);
fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white);
fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white);
fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white);
fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white);
draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0));
draw(arc((2,0),1,180,0)--(2,0)--cycle);
draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle);
draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle);
draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle);
draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle);
draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle);
label("$2$",(3.5,3sqrt(3)/2),NE);
[/asy]](//latex.artofproblemsolving.com/c/c/4/cc4a8eea36390b30289f00cdba179c25b9ee3543.png)
so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?![[asy]
size(140);
fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4));
fill(arc((2,0),1,180,0)--(2,0)--cycle,white);
fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white);
fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white);
fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white);
fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white);
fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white);
draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0));
draw(arc((2,0),1,180,0)--(2,0)--cycle);
draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle);
draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle);
draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle);
draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle);
draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle);
label("$2$",(3.5,3sqrt(3)/2),NE);
[/asy]](http://latex.artofproblemsolving.com/c/c/4/cc4a8eea36390b30289f00cdba179c25b9ee3543.png)
15
Steve wrote the digits , , , , and in order repeatedly from left to right, forming a list of 
digits, beginning 
He then erased every third digit from his list (that is, the rd, 
th, 
th, 
digits from the left), then erased every fourth digit from the resulting list (that is, the th, th, th, digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions 
?
, , , , and in order repeatedly from left to right, forming a list of 
digits, beginning 
He then erased every third digit from his list (that is, the rd, 
th, 
th, 
digits from the left), then erased every fourth digit from the resulting list (that is, the th, th, th, digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions 
?
16
Bela and Jenn play the following game on the closed interval ![$[0, n]$](//latex.artofproblemsolving.com/a/5/8/a585433224883cbcd304f34129a152b97d5ed96e.png)
of the real number line, where is a fixed integer greater than . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
![$[0, n]$](http://latex.artofproblemsolving.com/a/5/8/a585433224883cbcd304f34129a152b97d5ed96e.png)
of the real number line, where is a fixed integer greater than . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
17
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
19
In a certain card game, a player is dealt a hand of cards from a deck of 
distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 
. What is the digit 
?
cards from a deck of 
distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 
. What is the digit 
?
20
Let 
be a right rectangular prism (box) with edges lengths 
and , together with its interior. For real 
, let 
be the set of points in -dimensional space that lie within a distance 
of some point . The volume of can be expressed as 
, where 
and are positive real numbers. What is 
be a right rectangular prism (box) with edges lengths 
and , together with its interior. For real 
, let 
be the set of points in -dimensional space that lie within a distance 
of some point . The volume of can be expressed as 
, where 
and are positive real numbers. What is 
21
In square , points and 
lie on 
and 
, respectively, so that 
Points 
and 
lie on 
and 
, respectively, and points 
and 
lie on 
so that 
and 
. See the figure below. Triangle 
, quadrilateral 
, quadrilateral 
, and pentagon 
each has area 
What is 
?
![[asy]
real x=2sqrt(2);
real y=2sqrt(16-8sqrt(2))-4+2sqrt(2);
real z=2sqrt(8-4sqrt(2));
pair A, B, C, D, E, F, G, H, I, J;
A = (0,0);
B = (4,0);
C = (4,4);
D = (0,4);
E = (x,0);
F = (4,y);
G = (y,4);
H = (0,x);
I = F + z * dir(225);
J = G + z * dir(225);
draw(A--B--C--D--A);
draw(H--E);
draw(J--G^^F--I);
draw(rightanglemark(G, J, I), linewidth(.5));
draw(rightanglemark(F, I, E), linewidth(.5));
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, dir(90));
dot("$D$", D, dir(90));
dot("$E$", E, S);
dot("$F$", F, dir(0));
dot("$G$", G, N);
dot("$H$", H, W);
dot("$I$", I, SW);
dot("$J$", J, SW);
[/asy]](//latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png)
, points and 
lie on 
and 
, respectively, so that 
Points 
and 
lie on 
and 
, respectively, and points 
and 
lie on 
so that 
and 
. See the figure below. Triangle 
, quadrilateral 
, quadrilateral 
, and pentagon 
each has area 
What is 
?![[asy]
real x=2sqrt(2);
real y=2sqrt(16-8sqrt(2))-4+2sqrt(2);
real z=2sqrt(8-4sqrt(2));
pair A, B, C, D, E, F, G, H, I, J;
A = (0,0);
B = (4,0);
C = (4,4);
D = (0,4);
E = (x,0);
F = (4,y);
G = (y,4);
H = (0,x);
I = F + z * dir(225);
J = G + z * dir(225);
draw(A--B--C--D--A);
draw(H--E);
draw(J--G^^F--I);
draw(rightanglemark(G, J, I), linewidth(.5));
draw(rightanglemark(F, I, E), linewidth(.5));
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, dir(90));
dot("$D$", D, dir(90));
dot("$E$", E, S);
dot("$F$", F, dir(0));
dot("$G$", G, N);
dot("$H$", H, W);
dot("$I$", I, SW);
dot("$J$", J, SW);
[/asy]](http://latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png)
is divided by 
?
23
Square in the coordinate plane has vertices at the points 
and 
Consider the following four transformations:
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying and then 
would send the vertex at 
to 
and would send the vertex at 
to itself. How many sequences of 
transformations chosen from 
will send all of the labeled vertices back to their original positions? (For example, 
is one sequence of transformations that will send the vertices back to their original positions.)
in the coordinate plane has vertices at the points 
and 
Consider the following four transformations:
a rotation of counterclockwise around the origin;
a rotation of clockwise around the origin;
a reflection across the -axis; and
a reflection across the -axis.
a rotation of 
a rotation of 
a reflection across the 
a reflection across the Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying
and then 
would send the vertex at 
to 
and would send the vertex at 
to itself. How many sequences of 
transformations chosen from 
will send all of the labeled vertices back to their original positions? (For example, 
is one sequence of transformations that will send the vertices back to their original positions.)
24
How many positive integers satisfy
(Recall that 
is the greatest integer not exceeding .)
satisfy
(Recall that 
is the greatest integer not exceeding .)
25
Let 
denote the number of ways of writing the positive integer as a product
where 
, the 
are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number can be written as , 
, and 
, so 
. What is 
?
denote the number of ways of writing the positive integer as a product
where 
, the 
are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number can be written as , 
, and 
, so 
. What is 
?
These problems are copyright 
