Question 1

Leah has coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?

Solution

Question 2

What is ?

Solution

Question 3

Peter drove the first third of his trip on a gravel road, the next miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip?

Solution

Question 4

Susie pays for muffins and bananas. Calvin spends twice as much paying for muffins and bananas. A muffin is how many times as expensive as a banana?

Solution

Question 5

Camden constructs a square window using equal-size panes of glass, as shown. The ratio of the height to width for each pane is , and the borders around and between the panes are inches wide. In inches, what is the side length of the square window?

Solution

Question 6

Orvin went to the store with just enough money to buy balloons. When he arrived, he discovered that the store had a special sale on balloons: buy balloon at the regular price and get a second at off the regular price. What is the greatest number of balloons Orvin could buy?

Solution

Question 7

Suppose and is greater than . What is ?

Solution

Question 8

A truck travels feet every seconds. There are feet in a yard. How many yards does the truck travel in minutes?

Solution

Question 9

For real numbers and , What is ?

Solution

Question 10

In the addition shown below and are distinct digits. How many different values are possible for ?

Solution

Question 11

For the consumer, a single discount of is more advantageous than any of the following discounts:

(1) two successive discounts

(2) three successive discounts

(3) a discount followed by a discount

What is the smallest possible positive integer value of ?

Solution

Question 12

The largest divisor of is itself. What is its fifth largest divisor?

Solution

Question 13

Six regular hexagons surround a regular hexagon of side length as shown. What is the area of ?

Solution

We note that the 6 triangular sections in triangle ABC can be put together to form a hexagon congruent to each of the seven other hexagons (In the diagram I draw, the area of yellow triangle is same as 3 side triangles combined). By the formula for the area of the hexagon, we get the area for each hexagon as 3*sqrt(3)/2. The area of triangle ABC, which is equivalent to two of these hexagons together, is 3*sqrt(3).

Question 14

Danica drove her new car on a trip for a whole number of hours, averaging miles per hour. At the beginning of the trip, miles was displayed on the odometer, where is a 3-digit number with and . At the end of the trip, the odometer showed miles. What is  ?

Solution

Question 15

In rectangle , and points and lie on so that and trisect as shown. What is the ratio of the area of to the area of rectangle ?

Solution

Question 16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

Solution

Question 17

What is the greatest power of that is a factor of ?

Solution

Question 18

A list of positive integers has a mean of , a median of , and a unique mode of . What is the largest possible value of an integer in the list?

Solution

Question 19

Two concentric circles have radii and . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

Solution

Question 20

For how many integers is the number negative?

Solution

Question 21

Trapezoid has parallel sides of length and of length . The other two sides are of lengths and . The angles at and are acute. What is the length of the shorter diagonal of ?

Solution

Question 22

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

Solution

Question 23

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

Solution

First, we draw the vertical cross-section passing through the middle of the frustum.

Let the top base have a diameter of 2 (radius of 1), and the bottom base have a diameter of 2r (radius of r). Obviously the question is to ask for the value of r.

Using the Pythagorean theorem we have: (r+1)^2 = (2s)^2 + (r-1)^2, which is equivalent to: r^2+2r+1=4s^2+r^2-2r+1. Subtracting r^2-2r+1 from both sides, 4r=4s^2, or s = sqrt(r).

V_frustum = 1/3 * pi*h * (R^2 + r^2 + R*r)

Plug in the values of the question, we have:

V_frustum = 1/3 * pi*2s * (r^2 + 1^2 + r*1) = 1/3 * pi*2*sqrt(r) * (r^2 + r + 1)

V_sphere = 4/3 * pi * r^3

Plug in the values of the question, we have:

V_sphere = 4/3 * pi * s^3 = 4/3 * pi * sqrt(r)^3

We know V_frustum = 2 * V_sphere

So:
1/3 * pi*2*sqrt(r) * (r^2 + r + 1) = 2 * 4/3 * pi * sqrt(r)^3

r^2+r+1 = 4r
r^2-3r+1 = 0

r = (3+sqrt(5))/2

Question 24

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every from to one can find a subset of the numbers that appear consecutively on the circle that sum to . Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

Solution

no matter how you arrange the numbers, we can see that by choosing the full circle, we can obtain 15.
We can see that by choosing one number (1,2,3,4, or 5), we can always obtain subsets with sums 1, 2, 3, 4, and 5.
Also:
By choosing everything except for 1, we can get subset with sum of 14;
By choosing everything except for 2, we can get subset with sum of 13;
By choosing everything except for 3, we can get subset with sum of 12;
By choosing everything except for 4, we can get subset with sum of 11;
By choosing everything except for 5, we can get subset with sum of 10.

So this means that we now only need to check for 6, 7, 8, and 9. However, once we have found a set summing to 6, we can choose everything else and obtain a set summing to 9, and similarly for 7 and 8. Thus, we only need to check each case for whether or not we can obtain 6 or 7.

There is only 1 case, as shown below, where we cannot get a subset summing to 6, and only 1 case where we cannot get a subset summing to 7.

Question 25

In a small pond there are eleven lily pads in a row labeled through . A frog is sitting on pad . When the frog is on pad , , it will jump to pad with probability and to pad with probability . Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?

Solution

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