Type 1 Olympiad
Type 2 Olympiad
Type 3 Olympiad
Type 4 Olympiad
100

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are  cm in diameter and  cm high. Felicia buys cat food in cylindrical cans that are  cm in diameter and  cm high. What is the ratio of the volume of one of Alex's cans to the volume one of Felicia's cans?

Using the formula for the volume of a cylinder, we get Alex, , and Felicia, . We can quickly notice that  cancels out on both sides, and that Alex's volume is  of Felicia's leaving  as the answer.

100

Shauna takes five tests, each worth a maximum of  points. Her scores on the first three tests are , , and . In order to average  for all five tests, what is the lowest score she could earn on one of the other two tests?



100

Ike and Mike go into a sandwich shop with a total of  to spend. Sandwiches cost  each and soft drinks cost  each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?

 

100

The eighth grade class at Lincoln Middle School has  students. Each student takes a math class or a foreign language class or both. There are  eighth graders taking a math class, and there are  eight graders taking a foreign language class. How many eight graders take only a math class and not a foreign language class?

By P-I-E, we get  = .

Solving gives us .

But we want the number of students taking only a math class,

which is A. .

200

Easy Easy

Point (7,2) is translated up right 2 and down 5 units.

what is the new coordinate

(9,-3)

200

3x3-2x2-12x+8


Answer in factored form

(x+2)(x-2)(3x-2)

200

Carrie has a rectangular garden that measures  feet by  feet. She plants the entire garden with strawberry plants. Carrie is able to plant  strawberry plants per square foot, and she harvests an average of  strawberries per plant. How many strawberries can she expect to harvest?

D.


The area of the garden is  square feet. Since Carrie plants  strawberry plants per square foot, there are a total of  strawberry plants, each of which produces  strawberries on average. Accordingly, she can expect to harvest  strawberries.


200

How many integers between  and  have four distinct digits arranged in increasing order? (For example,  is one integer.)

C.


As in Solution 1, we find that the first two digits must be , and the third digit must be at least . If it is , then there are  choices for the last digit, namely , , , , or . Similarly, if the third digit is , there are  choices for the last digit, namely , , , and ; if , there are  choices; if , there are  choices; and if , there is  choice. It follows that the total number of such integers is

300

Ricardo has  coins, some of which are pennies (-cent coins) and the rest of which are nickels (-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?


C.



300

For a positive integer  the factorial notation  represents the product of the integers from  to . (For example, .) What value of  satisfies the following equation?

A.


We have , and . Therefore, the equation becomes , and so . Cancelling the s, it is clear that .

300

What does (sin2 θ + cosθ)=?

1

300

How many factors of  have more than  factors? (As an example,  has  factors, namely  and )

7

Since , we can simply list its factors:There are  of these; only  (i.e.  of them) don't have over  factors, so the remaining  factors have more than  factors.

400

cos 45 +sin 60=? 


Hint: Evaluate the trig functions and combine them

Answer in simplest radical form

√2/2 + √3/2

400

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

B. 4

400

Consider these two operations:

What is the value of

We can substitute , , and  into the functions' :

400

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned  years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is  years. How many years older than Bella is Anna?

Five years ago, Bella was  years old, and the kitten was  years old.

Today, Bella is  years old, and the kitten is  years old. It follows that Anna is  years old.

Therefore, Anna is  years older than Bella.

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