The rectangular floor of a bathroom is covered with square tiles (all of the same size). A spider starts at one corner of the bathroom and walks to the diagonally opposite corner. For example, the figure below shows a 6×8 bathroom, in which the spider touches 12 tiles on its path. (A spider doesn't touch a tile if it just walks over the grout at the corner of a tile.) For an m×n bathroom, how many tiles does the spider touch on its walk?
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square P in the bottom row and square Q in the top row. A marker is placed at P. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there from P to Q? (The figure shows a sample path.)
In triangle ABC, point D divides side AC so that AD:DC=1:2. Let E be the midpoint of BD and let F be the point of intersection of line BC and line AE. Given that the area of △ABC is 360, what is the area of △EBF?
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at 1:30 traveling due north at a steady 8 miles per hour. Betsy leaves on her bicycle from the same point at 2:30, traveling due east at a steady 12 miles per hour. At what time will they be exactly the same distance from their common starting point?
A box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second nut mix containing 20 percent peanuts, 40 percent cashews, and 40 percent almonds is added to the box resulting in a new nut mix that is 40 percent peanuts. How many pounds of cashews are now in the box?
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is 15. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from 12 to 14. If Ash plays with the teachers, the average age on that team will decrease from 55 to 52. How old is Ash?
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
Suppose a and b are real numbers. When the polynomial x3+x2+ax+b is divided by x−1, the remainder is 4. When the polynomial is divided by x−2, the remainder is 6. What is b−a?
Agnes writes the following four statements on a blank piece of paper.
• At least one of these statements is true. • At least two of these statements are true. • At least two of these statements are false. • At least one of these statements is false.
Each statement is either true or false. How many false statements did Agnes write on the paper?
A semicircle has diameter AB and chord CD of length 16 parallel to AB. A smaller semicircle with diameter on AB and tangent to CD is cut from the larger semicircle, as shown below. What is the area of the resulting figure, shown shaded?
The sequence 1,x,y,z is arithmetic. The sequence 1,p,q,z is geometric. Both sequences are strictly increasing and contain only integers, and z is as small as possible. What is the value of x+y+z+p+q?
Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is 0. How many 4-digit passcodes satisfy these conditions?
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is k, where 0<k<1. The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale).
The area of the shaded portion of the figure is 64% of the area of the original square. What is k?
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?
Let N be the unique positive integer such that dividing 273436 by N leaves a remainder of 16 and dividing 272760 by N leaves a remainder of 15. What is the tens digit of N?
An array of numbers is constructed beginning with the numbers −1, 3, and 1 in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with −1 and 1, respectively. If the process continues, one of the rows will sum to 12,288. In that row, what is the third number from the left?
A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and g>0 meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of g can be written as dab−c, where a, b, c, and d are positive integers, b is not divisible by the square of any prime, and d is relatively prime to the greatest common divisor of a and c. What is a+b+c+d?
A set of numbers is called sum-free if whenever x and y are (not necessarily distinct) elements of the set, x+y is not an element of the set. For example, {1,4,6} and the empty set are sum-free, but {2,4,5} is not. What is the greatest possible number of elements in a sum-free subset of {1,2,3,…,20}?
A circle of radius r is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown below. What is r?
Call a positive integer fair if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater digits. For example, 196, 23, and 12463 are fair, but 1546, 320, and 34321 are not fair. How many fair positive integers are there?
A point P is chosen at random inside square ABCD. The probability that AP is neither the shortest nor the longest side of △APB can be written as ea+bπ−cd, where a,b,c,d,e are positive integers, gcd(a,b,c,e)=1, and d is not divisible by the square of a prime. What is a+b+c+d+e?
In a certain alien world, the maximum running speed v of an organism is dependent on its number of toes n and number of eyes m. The relationship can be expressed as v=knamb centimeters per hour, where k, a, and b are integer constants. In a population where all organisms have 5 toes, logv=4+2logm; and in a population where all organisms have 25 eyes, logv=4+4logn, where the logarithms are base 10. What is k+a+b?
In the figure shown below, major arc AD and minor arc BC have the same center, O. Also, A lies between O and B, and D lies between O and C. Major arc AD, minor arc BC, and each of the two segments AB and CD have length 2π. What is the distance from O to A?
The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are A(2,31), B(8,27), and C(18,27)?
Let C={1,2,3,…,13}. Let N be the greatest integer such that there exists a subset of C with N elements that does not contain five consecutive integers. Suppose N integers are chosen at random from C without replacement. What is the probability that the chosen elements do not include five consecutive integers?
Points F, G, and H are collinear with G between F and H. The ellipse with foci at G and H is internally tangent to the ellipse with foci at F and G, as shown below. The two ellipses have the same eccentricity e, and the ratio of their areas is 2025. (Recall that the eccentricity of an ellipse is e=ac, where c is the distance from the center to a focus, and 2a is the length of the major axis.) What is e?
The base of the pentahedron shown below is a 13×8 rectangle, and its lateral faces are two isosceles triangles with base of length 8 and congruent sides of length 13, and two isosceles trapezoids with bases of length 7 and 13 and nonparallel sides of length 13. What is the volume of the pentahedron?
Three real numbers are chosen independently and uniformly at random between 0 and 1. What is the probability that the greatest of these three numbers is greater than 2 times each of the other two numbers? (In other words, if the chosen numbers are a≥b≥c, then a>2b.)
A circle of radius r is surrounded by 12 circles of radius 1, externally tangent to the central circle and sequentially tangent to each other, as shown. Then r can be written as a+b+c, where a,b,c are integers. What is a+b+c?
Polynomials P(x) and Q(x) each have degree 3 and leading coefficient 1, and their roots are all elements of {1,2,3,4,5}. The function f(x)=Q(x)P(x) has the property that there exist real numbers a<b<c<d such that the set of all real numbers x such that f(x)≤0 consists of the closed interval [a,b] together with the open interval (c,d). How many functions f(x) are possible?
The instructions on a 350-gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires 20 grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?
A Pascal-like triangle has 10 as the top row and 10 followed by 1 as the second row. In each subsequent row the first number is 10, the last number is 1, and, as in the standard Pascal's Triangle, each other number in the row is the sum of the two numbers directly above it. The first four rows are shown below: 10101021101111211 What is the sum of the digits of the sum of the numbers in the 11th row?
The line y=31x+1 divides the square region defined by 0≤x≤2 and 0≤y≤2 into an upper and lower region. The line x=a divides the lower region into two regions of equal area. Then a can be written as s−t, where s and t are positive integers. What is s+t?
Frances stands 15 meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located x meters east of the locked gate. An unlocked gate lies 9 meters east of the box, and another unlocked gate lies 8 meters west of the locked gate. Frances can reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel is the same via either unlocked gate. What is the value of x?
Emmy says to Max, "I ordered 36 math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was ABB.BA, where A and B are digits and A=0." After a pause, Max says, "That was a good price." What is A+B?
