Puzzles
Two semicircles
The diagram shows two semicircles.
![](javascript:showlimage("two-semicircles.jpg"))
\(CD\) is a chord of the larger circle and is parallel to \(AB\). The length of \(CD\) is 8m. What is the area of the shaded region (in terms of \(\pi\))?
What is the sum?
What is \(\displaystyle\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{15}+\sqrt{16}}\)?
Between quadratics
Source: Luciano Rila (@DrTrapezio)
\(p(x)\) is a quadratic polynomial with real coefficients. For all real numbers \(x\),
$$x^2-2x+2\leq p(x)\leq 2x^2-4x+3$$
\(p(11)=181\). Find \(p(16)\).
24 December
Today's number is the largest possible remainder which can be obtained when dividing one of
the answers in this advent calendar by another answer smaller than it (not
including today's answer!).
23 December
This number is a prime number. If you treble it and add 16, the result is also prime. Repeating this will give 11 prime numbers in total (including the number itself).
22 December
What is the largest number which cannot be written as the sum of distinct squares?
21 December
This year, I posted instructions for making a dodecahedron and a stellated rhombicuboctahedron.
To get today's number, multiply the number of modules needed to make a dodecahedron by half the number of tube maps used to make a stellated rhombicuboctahedron.
20 December
Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums reading across and down are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10.
| + | - | = 8 | |||
| - | - | - | |||
| + | ÷ | = 9 | |||
| + | ÷ | × | |||
| + | × | = 108 | |||
| = 6 | = 1 | = 18 |
The answer is the product of the digits in the red boxes.