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### Unleash Your Inner Rectangle

The midpoints of the sides of quadrilateral *ABCD* form the vertices of a rectangle. Angle *CAB* is 17 degrees.

* What is the measure of angle ABD?*

### Pent-up-gon

The midpoints of the sides of pentagon *ABCDE* form the vertices of a regular pentagon.

*What is the length of side BC divided by the length of side CD?*

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## Solutions to week 75

**Some Difference**. First, let’s find the smallest and largest birthday sums to get an idea of how big the difference could possibly be. We’ll write dates like 68-2-18 for the example birthday, Feb 18, 1968, so that you can just add up the three numbers to get the birthday sum. The smallest possible birthday sum occurs for someone born on 00-1-1, namely 2; we find this by making each of the components as small as possible. Making each of the components as large as possible, the largest birthday sum, 142, goes to someone born on 99-12-31. But notice that 00-1-1 is the day after 99-12-31. So the largest possible difference of birthday sums of people born on consecutive days is **140**.

**In Your Prime**. Note that after your first decade, all prime ages are odd, and 10*k*+5 is never prime (always divisible by five), so the only possibility for the *k+1*st decade to be prime is if all of 10*k*+1, 10*k*+3, 10*k*+7, and 10*k*+9 are prime. However, if *k* is a multiple of three, then 10*k*+3 is also a multiple of three, so that won’t be a prime decade. Similarly, if *k* is one less than a multiple of three, say 3*j*-1, then 10*k*+1 = 30*j*-10+1 = 30*j*-9 is a multiple of three. So only a decade for which *k* is one *more* than a multiple of three could work. We first try the forties, but 49 is not prime. Then we try the seventies, but 77 is not prime. The next possibility is the first decade after turning 100, and indeed, all of 101, 103, 107, and 109 are prime. Hence, your first prime decade after your teen years is your **eleventh** decade. Mathematically speaking, it’s clearly good to be a centenarian!

## Recent Weeks

**Week 75**: Some Difference & In Your Prime, solutions to Frequent Figures & Doubled Digit

**Week 74**: Frequent Figures & Doubled Digit, solutions to Small Sudoku & Puny Pegs

**Week 73**: Small Sudoku & Puny Pegs, solutions to The Second Bisection & The Third Circle

**Week 72**: The Second Bisection & The Third Circle, solutions to Eye Like It & Pythagorean Poster

**Week 71**: Eye Like It & Pythagorean Poster, solutions to In The Groove & On The Path

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.

[asciimathsf]