vmt-award-2024_-46-2-_yellow

Solutions to 46(1) HEXA Challenge Problems

November Challenge : In a room of 150 people, each person shakes the hand of each other person one. How many different handshakes are shared? SOLUTION: The first person shakes 150 - 1 hands; the second person shakes 150 - 2 hands; the third person shakes 150 - 3 hands; and so forth. In total, there are (150 - 1) + (150 - 2) + (150 - 3) . . . + 2 + 1 hands shake shared, such that (150 - 1) + (150 - 2) + (150 - 3) + … + (150 - 75) +1 +2 + … +74 which gives us (150 - 1) handshakes 75 times. Thus, we have (150 - 1) × 75 = 11,175 handshakes. December Challenge: There are 5 people born on January 1st, but in 5 consecutive leap years. The last known leap year was 2016. the two youngest are 42 years old together. What year was the oldest person born (given that we are in 2019)?

SOLUTION: Let the youngest person be x years old. The second youngest person is x + 4 years old. Given that the sum of their ages is 42 years, we find the youngest person ’ s age as follows:

x + x + 4 = 42 2 x + 4 = 42 2 x = 38 x = 19

The youngest person is 19 years old. The oldest person is 16 years older than the youngest of 19 + 16 = 35 years old. The oldest person was born in the year 1984.

January Challenge : Ioana picks out two numbers a and b from the set {1, 2, 3, …, 26}. The product ab is equal to the sum of the re maining 24 numbers. What is the value of | a - b |

SOLUTION: Assume that a ≠ b . The sum of the set is 351 such that ab = 351 - a - b .

As the range of values for ab can be found by letting a and b equal the smallest possible integers from the set (1 and 2, respectably) and then letting a and b equal the largest possible integers from the set (25 and 26, respective ly) such that 2 ≤ ab 650 and 300 ≤ 351 - a - b ≤ 348

Since ab = 351 - a - b , we can substitute and narrow the range such that

300 ≤ ab ≤ 348

Now, we can eliminate values of a and b that do not fall into this range. If b = 26, then a must be greater than 11, which eliminates integers 1 - 11 for values of both a and b .

Virginia Mathematics Teacher vol. 46, no. 2

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