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Induction HM WK

The document contains a series of math exercises and problems for students preparing for competitions, including warm-up questions and challenge problems. It emphasizes the use of mathematical induction in some exercises and provides specific problems from various Streamline Olympiads. The problems range in difficulty and cover topics such as digit sums, properties of integers, and coin weighing puzzles.

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sadikov.kamal99
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26 views2 pages

Induction HM WK

The document contains a series of math exercises and problems for students preparing for competitions, including warm-up questions and challenge problems. It emphasizes the use of mathematical induction in some exercises and provides specific problems from various Streamline Olympiads. The problems range in difficulty and cover topics such as digit sums, properties of integers, and coin weighing puzzles.

Uploaded by

sadikov.kamal99
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Induction.

Other Homework problems


Tanya Khovanova
May 4, 2009

“The problems for the exam will be similar to the ones discussed in class.
Of course, the numbers will be different. But not all of them. Pi will still be
3.14159... ”

Warm-Up
Exercise 1. Humans have 10 fingers on their hands. How many fingers are
there on 10 hands?

Competition practice
Exercise 2. 2000 Streamline Olympiad. 6th-7th grade. Let A be the
least integer such that the sum of all its digits is equal to 2000. Find the
left-most digit of A.
Exercise 3. 1998 Streamline Olympiad. 8th-9th grade. Find three
numbers such that each of them is a square of the difference of the two others.
Exercise 4. 1999 Streamline Olympiad. 9th-10th grade. The positive
integers 30, 72, and N have a property that the product of any two of them
is divisible by the third. What is the smallest possible value of N ?
Exercise 5. 1999 Streamline Olympiad. 9th-10th grade. You have 6
coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same. The number (1,
2, 3, 4, 5, 6) on the top of each coin should correspond to its weight. How
can you determine whether all the numbers are correct, using the balance
scale only twice?

1
Exercise 6. 2000 Streamline Olympiad. 8th grade. You have six bags
with coins that look the same. Each bag has an infinite number of coins and
all coins in the same bag weigh the same amount. Coins in different bags
weigh 1, 2, 3, 4, 5 and 6 grams exactly. There is a label (1, 2, 3, 4, 5, 6)
attached to each bag that is supposed to correspond to the weight of the
coins in that bag. You have only a balance scale. What is the least number
of times do you need to weigh coins in order to confirm that the labels are
correct?

Challenge Problems
Exercise 7. Use mathematical induction to prove that given n straight lines
on a plane you can color the regions created by the lines into red and blue
in such a way that any two neighboring regions are colored with different
colors.

Exercise 8. Use induction to prove that if x+1/x is an integer then xn +1/xn


is also an integer.

Exercise 9. Prove that the second to last digit of every power of 3 is even.

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