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Problem Solving
1. Call a set of positive numbers a “phancy set” if the product of any two integers in the set is
one less than a perfect square. What is the least possible value for n such that {4, 6, n} is a
phancy set?
2. Three 3-digit numbers are formed using the digits 1 through 9 exactly once each. The
hundreds digit of the first number is 1. The tens digit of the second number is 8. The
units digit of the third number is 5. The ratio of the first number to the second number to
the third number is 1:3:5, respectively. What is the sum of the three numbers?
3. Find the largest integer less than 1000 which leaves remainders of 1 when divided by 2, 3, 5,
and 7.
4. Each of the figures is a square formed by connecting
midpoints of opposite sides of a larger square. What
fraction of the largest square is shaded?
5. The numbers 1, 2, 3, 4, 5, and 6 are placed in the six circles
shown to the right so that the sum along each edge of the
triangle is the same. What is the smallest possible sum along
an edge?
6. In magic squares, the sum of the numbers in each
row, each column, and each diagonal is constant.
Find the value of B in the following magic square.
7. Place the numbers {1,2,3,4,5,6,7,8,9} in each of the nine
smaller squares to the right to create a magic square.