- #1
Ben1587
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These we're the problems that I've tried to do. Solutions with steps would greatly be appreciated.
1. Suppose that a, b, c, are distinct non-zero digits.
(A) Find a formula, depending on a,b, c, for the sum of all six digit integers whose only digits are a,b,c.
(B) What is the sum of all six digit integers having no non-zero digits?
2. Verify that 9^2+2^2=7^2+6^2 by cutting a 9x9 square into four pieces that together with a 2x2 square can be rearranged to give a 7x7 square and a 6x6 square.
3. A point within an equilateral triangle is at distance 5, 7, 8 from each of the vertices. What is the length of a side of the triangle?
4. A photographer wished to arrange twelve people of different heights in two rows of six each. Each person in the first row must be shorter than the person directly behind and, going from left to right, the people must get taller.
How many such arrangements are there?
5. Let S be a set of positive integers, each of which is less than 20.
(A) What is the maximum size m of S if S has the property that no sum distinct elements of S is equal to 20.
(B) Of all sets S of positive integers less than 20 having maximal size m, and having the property that no sum of distinct elements of S is equal to 20, which are the sets having the samllest possible sum?
Thanks
1. Suppose that a, b, c, are distinct non-zero digits.
(A) Find a formula, depending on a,b, c, for the sum of all six digit integers whose only digits are a,b,c.
(B) What is the sum of all six digit integers having no non-zero digits?
2. Verify that 9^2+2^2=7^2+6^2 by cutting a 9x9 square into four pieces that together with a 2x2 square can be rearranged to give a 7x7 square and a 6x6 square.
3. A point within an equilateral triangle is at distance 5, 7, 8 from each of the vertices. What is the length of a side of the triangle?
4. A photographer wished to arrange twelve people of different heights in two rows of six each. Each person in the first row must be shorter than the person directly behind and, going from left to right, the people must get taller.
How many such arrangements are there?
5. Let S be a set of positive integers, each of which is less than 20.
(A) What is the maximum size m of S if S has the property that no sum distinct elements of S is equal to 20.
(B) Of all sets S of positive integers less than 20 having maximal size m, and having the property that no sum of distinct elements of S is equal to 20, which are the sets having the samllest possible sum?
Thanks