The Lion

The LionTeam Competition

• Welcome to the team competition. This is the event where using all your teammates to optimize your time will be essential.

• Each question will have a different allotment of time and each question varies in difficulty.

• For each question you may score 2 points. One point for a speed answer and a second point for your final answer. (Note: to get the speed answer point it must be same correct answer as your final answer.)

• Let’s do a practice question!

Speed answer: 2 minutes

Final Answer: 3 minutes

Practice question time:

Practice questionIn Circle Land, all rules of mathematics are the same as we know them except numbers are shown in the following way:

Calculate the following expression and provide the answer as they would in Circle Land?

Practice questionIn Circle Land, all rules of mathematics are the same as we know them except numbers are shown in the following way:

Calculate the following expression and provide the answer as they would in Circle Land?

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Practice question answer

Now for the real thing. Good luck.

Question #1 time:

In the diagram, ABCD is a square with area 25 cm2 . If PQCD is a rhombus with area 20 cm2.

What is the area of the shaded region?

Question #1

In the diagram, ABCD is a square with area 25 cm2 . If PQCD is a rhombus with area 20 cm2.

What is the area of the shaded region?

Question #1

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Speed answer: 4.5 minutes

Question #2 time:

Question #2A hat contains n slips of paper. The slips of paper are numbered with consecutive even integers from 2 to 2n. Consider the situation where there are six slips of paper (n = 6), in the hat, two students, Tarang and Joon, will each choose three slips from the hat and sum their total. In this situation (when n=6) it is impossible for them to have the same total.

If more slips of paper are added to the hat, what is the smallest value of n > 6 so that Tarang and Joon can each choose half of the slips and obtain the same total?

Question #2A hat contains n slips of paper. The slips of paper are numbered with consecutive even integers from 2 to 2n. Consider the situation where there are six slips of paper (n = 6), in the hat, two students, Tarang and Joon, will each choose three slips from the hat and sum their total. In this situation (when n=6) it is impossible for them to have the same total.

If more slips of paper are added to the hat, what is the smallest value of n > 6 so that Tarang and Joon can each choose half of the slips and obtain the same total?

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Question #3 time:

Question #3The Fryer Foundation is giving out four types of prizes, valued at $5, $25, $125 and $625.

There are two ways in which the Foundation could give away prizes totaling $880 while making sure to give away at least one and at most six of each prize. Determine the two ways this can be done.

Question #3The Fryer Foundation is giving out four types of prizes, valued at $5, $25, $125 and $625.

There are two ways in which the Foundation could give away prizes totaling $880 while making sure to give away at least one and at most six of each prize. Determine the two ways this can be done.

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Question #4 time:

Question #4

A Nakamoto triangle is a right-angled triangle with integer side lengths which are in the ratio 3 : 4 : 5. (For example, a triangle with side lengths 9, 12 and 15 is a Nakamoto triangle.) There are three Nakamoto triangles that have a side length of 60. Find the combined area of these three triangles.

Question #4

A Nakamoto triangle is a right-angled triangle with integer side lengths which are in the ratio 3 : 4 : 5. (For example, a triangle with side lengths 9, 12 and 15 is a Nakamoto triangle.) There are three Nakamoto triangles that have a side length of 60. Find the combined area of these three triangles.

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Question #5 time:

Question #5• Let s be the number of positive integers from 1 to 100, inclusive, that do not contain the digit 7.

• Let t be the number of positive integers from 101 to 300, inclusive, that do not contain the digit 2.

• Let f be the number of positive integers from 3901 to 5000, inclusive, that do not contain the digit 4.

Determine the value of s+t+f.

Question #5• Let s be the number of positive integers from 1 to 100, inclusive, that do not contain the digit 7.

• Let t be the number of positive integers from 101 to 300, inclusive, that do not contain the digit 2.

• Let f be the number of positive integers from 3901 to 5000, inclusive, that do not contain the digit 4.

Determine the value of s+t+f.

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Question #6 time:

Question #6Amanda has the following grades on her Calculus tests this past year. She can’t remember her grade on her Parametrics test but she does remember that her worst test was Applications of Integration. She took her Polar test today but has no idea how she did on it. What is the difference between her highest possible average for the year and her lowest possible average?

Limits 98

Derivatives 80

Applications of Derivatives

Integration 85

Applications of Integration

Integration Techniques

Series 91

Parametrics ???

Polar ???

Question #6Amanda has the following grades on her Calculus tests this past year. She can’t remember her grade on her Parametrics test but she does remember that her worst test was Applications of Integration. She took her Polar test today but has no idea how she did on it. What is the difference between her highest possible average for the year and her lowest possible average?

Limits 98

Derivatives 80

Applications of Derivatives

Integration 85

Applications of Integration

Integration Techniques

Series 91

Parametrics ???

Polar ???

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Question #7 time:

Question #7The odd positive integers are arranged in rows in the triangular pattern, as shown. Determine the row where the number 1001 occurs.

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Question #8 time:

Question #8Dmitri has a collection of identical cubes. Each cube is labeled with the integers 1 to 6 as shown in the following net: (This net can be folded to make a cube.)

He forms a pyramid by stacking layers of the cubes on a table, as shown, with the bottom layer being a 7 by 7 square of cubes.

• Let a be the total number of blocks used.• When all the visible numbers are added up, let b be the smallest possible total.• When all the visible numbers are added from a bird’s eye view, let c be the largest possible sum.

Determine the value of a+b+c.

Question #8Dmitri has a collection of identical cubes. Each cube is labeled with the integers 1 to 6 as shown in the following net: (This net can be folded to make a cube.)

He forms a pyramid by stacking layers of the cubes on a table, as shown, with the bottom layer being a 7 by 7 square of cubes.

• Let a be the total number of blocks used.• When all the visible numbers are added up, let b be the smallest possible total.• When all the visible numbers are added from a bird’s eye view, let c be the largest possible sum.

Determine the value of a+b+c.

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Question #9 time:

A number is Beprisque if it is the only natural number between a prime number and a perfect square (e.g. 10 is Beprisque but 12 is not). Find the sum of the first five Beprisque numbers (including 10).

Question #9

A number is Beprisque if it is the only natural number between a prime number and a perfect square (e.g. 10 is Beprisque but 12 is not). Find the sum of the first five Beprisque numbers (including 10).

Question #9

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Question #10 time:

Question #10

Felix the cat, wants to give fresh fish to his girlfriend Kitty, as her birthday present. To do this Felix has to walk to the Lumba River, catch a bucket of fish and walk to Kitty’s house. The Lumba River is located 105m north of Felix’s house, and runs straight east and west. Kitty’s house is located 195m south of the Lumba River. If the distance between Felix’s house and Kitty’s house is 410m, what is the shortest route that Felix can take from his house to the river and finally to Kitty’s house?

Felix’s House

Kitty’s House

Fishing

Lumba River

Question #10

Felix the cat, wants to give fresh fish to his girlfriend Kitty, as her birthday present. To do this Felix has to walk to the Lumba River, catch a bucket of fish and walk to Kitty’s house. The Lumba River is located 105m north of Felix’s house, and runs straight east and west. Kitty’s house is located 195m south of the Lumba River. If the distance between Felix’s house and Kitty’s house is 410m, what is the shortest route that Felix can take from his house to the river and finally to Kitty’s house?

Felix’s House

Kitty’s House

Fishing

Lumba River

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Question #11 time:

Question #11In a large grassy field there is a rectangular barn of dimensions 11m x 5m. There are 2 horses and a llama tied to ropes, the ropes are attached to the barn at points A, B and C. The rope at point A is 7 m long, the rope at point B is 4 m long and the rope at point C is 5 m long. Based upon the length of the rope, in terms of determine the area of grass that the animals will be able to reach.

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Question #12 time:

Question #12A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2882 is a four-digit palindrome and 49194 is a five-digit palindrome. There are pairs of four digit palindromes whose sum is a five-digit palindrome. One such pair is 2882 and 9339. How many such pairs are there?

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Question #13 time:

Question #13•A “double-single” number is a three-digit number made up of two identical digits followed by a different digit. For example, 553 is a double-single number. Let d be the number of double-single numbers between 100 and 1000?

• Let s be the sum of the digits of the integer equal to 777 777 777 777 7772 - 222 222 222 222 2232

•Determine the difference between d and s.

Question #13•A “double-single” number is a three-digit number made up of two identical digits followed by a different digit. For example, 553 is a double-single number. Let d be the number of double-single numbers between 100 and 1000?

• Let s be the sum of the digits of the integer equal to 777 777 777 777 7772 - 222 222 222 222 2232

•Determine the difference between d and s.

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That’s all folks!

Practice QuestionSpeed Answer:

Team: _______

Practice QuestionFinal Answer:

Team: _______

Question #1Speed Answer:

Team: _______

Question #1Final Answer:

Team: _______

Question #3

Speed Answer:

Team: _______

$5 $25 $125 $625

Question #3

Final Answer:

Team: _______

$5 $25 $125 $625

Team: _______

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