Daily math problems for May 6, 2026 — Probability, Geometry, Number Theory

Compute the complement — probability all 23 birthdays are distinct: $$P(\text{all distinct})=\frac{365}{365}\cdot \frac{364}{365}\cdot \frac{363}{365}\cdots \frac{343}{365}\approx 0.493$$

So $P(\text{at least one match})=1-0.493\approx 0.507$, just over 50\%.

The diagonal of the inscribed square equals the circle's diameter, $2r$. For a square with diagonal $d$, the side length is $d\mathrm{/}\sqrt{2}$, so the area is $(d\mathrm{/}\sqrt{2}{)}^2=d^2\mathrm{/}2$.

Here $d=2r$, so area $=(2r{)}^2\mathrm{/}2=4r^2\mathrm{/}2=2r^2$.

Work modulo 5. Note $7\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}5)$, so $7^{100}\equiv 2^{100}(\mathrm{m}\mathrm{o}\mathrm{d}5)$.

Now look at powers of 2 mod 5: $2^1=2$, $2^2=4$, $2^3=8\equiv 3$, $2^4=16\equiv 1$. The cycle has length 4.

Since $100=4\cdot 25$, $2^{100}=(2^4{)}^{25}\equiv 1^{25}=1(\mathrm{m}\mathrm{o}\mathrm{d}5)$.