Olympiad Common Problems - 5th Grade

1). Problem type: Find missing number of a list given the average.
Example:
Julian has scored 93 and 96 on his first two tests. With one test remaining, what is the least score Julian needs to average 90 or better on his three tests? 		Solution:
1. Let N be the missing score.
2. The equation for the average is:
    (93 + 96 + N) / 3 = 90.
3. Solve this equation for N


    N = ____.
2). Problem type: Areas of composite shapes
Example:
How much carpeting is needed to cover the floor of an L shaped room with the dimensions shown in the figure?
 Solution:
1. Write in the dimensions of the missing sides on the diagram.
2. Cut the figure into two rectangles, cutting horizontally.
3. Find the area of each rectangle and add them together.
3.) Problem type: Arithmetic sequence:
Example:
Pictured is a sequence of growing chairs. The first chair is made of 6 squares. How many squares are in the 20th chair?
 Solution:
1. Under each chair write the number of squares in the chair.
2. Determine D, the number of squares added to each chair. D = ____
3. The equation for the nth term of an arithmetic sequence is:
    An = A1 + D (n - 1), where:
    A1 is the first term
    An is the nth term (the one you're looking for)
4. Write out the equation with the parameters of this problem and solve for A20



A20 = _____

4). Problem type: Guess-and-check
Example:
A farm keeps chickens (with 2 legs) and pigs (with 4 legs) in a pen. There are a total of 55 chickens and pigs in the pen. There are a total of 178 legs amongst them all. How many pigs are in the pen?
Solution:
Use this table: (make sure the number of heads adds to 55)
Guess ## Chickens# PigsTotal
heads		Total
feet
1302555160
     
     
     
     

Number of pigs = _____
5). Problem type: Permutations
Example:
Four students form a line to use a water fountain. How many different ways can they arrange themselves in line?
 Solution:
1. The number of ways to pick the first student is ____
2. After picking that student, how many students are left? ____
3. After picking that student, how many students are left? ____
4. How many are left after that pick? ___
5. Multiply these together to get the total number of ways to arrange the students = _____.
6.) Problem type: Working a problem backwards
Example:
Jason is playing with a stack of cards. He divides the cards into 3 equal piles. Then he takes one pile and divides it into 4 equal piles. Then he takes one of the four equally divided piles and further divides it into 5 equal piles. If one of the five equally divided piles contains 3 cards, how many cards in total are in Jason's stack?
 Solution:
1. If Jason has 5 piles at the end and each pile contains 3 cards then those 5 piles contain ____ cards total.
2. If those 5 piles were 1/4 of the cards in the previous step, then the 4 piles together contain ____ cards.
3. Those 4 piles are 1/3 of the deck, so the whole deck contained ____ cards

7). Problem type: Turning words into equations:
Example:
I'm thinking of a number. If I multiply my number by 12 and then add 2, I get the same result as if I had added 90 to my number. What is my number?
 Solution:
Using N for the number,
1. Write the expression for "multiply my number by 12 and then add 2" = ____________.
2. Write the expression for
"add 90 to my number" = _______.
3. Set these two expressions equal to each other and solve.
N = __________
8). Problem type: Cartesian coordinates
Example:
1) A parallelogram has vertices at
(0, 0), (5, 2), and (1, 3).
The fourth vertex has positive coordinates. What are they?
Solution:
l. Plot the 3 points on the graph.
2. Since the opposite sides of a parallelogram are parallel, the third vertex and the 4th vertex are parallel to the first and second.
3. The offset of the second vertex from the first is (___,___).
4. Apply this offset to the third vertex to find the 4th = (___,___).