Portia’s high school has 3 times as many students as Lara’s high school. The two high schools have a total of

2600 students. How many students does Portia’s high school have?

The sum of two natural numbers is 17402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time

interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the

bottom of the hill. How far, in inches, does it travel?

The quiz scores of a class with k>12 students have a mean of 8. The mean of a collection of 12 of these quiz scores is 14.  What is the mean of the remaining quiz scores in terms of k?

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2, the second sheet contains pages 3 and 4, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 19. How many sheets were borrowed?

Frieda the frog begins a sequence of hops on a 3 x 3 grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?

A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15?

A game board consists of 64 squares that alternate in color between black and white. The figure below shows square P in the bottom 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.)