Puzzles

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14 December

This puzzle was part of the 2023 Advent calendar.

All 2023 advent puzzles ~ More information

The function

f (x) = a x + b

(where

a

and

b

are real constants) satisfies

- x^{3} + 2 x^{2} + 6 x - 9 ⩽ f (x) ⩽ x^{2} - 2 x + 3

whenever

0 ⩽ x ⩽ 3

. What is

f (200)

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How to get started with this puzzle is easiest to see if we plot the two curves:

We see that the two curves meet at the point (2,3). The only way that the straight line

f

can be between the two curves is if it is a tangent to both curves.

By differentiation of another way of finding tangents, you can show that

a = 2

and

b = 1

. Therefore

f (200)

is 399.

Tags: algebra, functions, quadratics, cubics

Find them all

Source: Twenty-six Years of Problem Posing by John Mason

Find all continuous positive functions,

f

[0, 1]

such that:

\int_{0}^{1} f (x) d x = 1 and \int_{0}^{1} x f (x) d x = α and \int_{0}^{1} x^{2} f (x) d x = α^{2}

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0 = α^{2} - 2 α^{2} + α^{2} = \int_{0}^{1} x^{2} f (x) d x - 2 \int_{0}^{1} α x f (x) d x + \int_{0}^{1} α^{2} f (x) d x = \int_{0}^{1} (x^{2} - 2 α x + α^{2}) f (x) d x = \int_{0}^{1} (x - α)^{2} f (x) d x

f (x)

and

(x - α)^{2}

are both positive so this is only possible if one of them if always zero.

But

(x - α)^{2}

is only zero when

x = α

and

\int_{0}^{1} f (x) d x = 1

f (x)

cannot always be zero. Therefore no such function exists.

Extension

Find all continuous positive functions,

f

[0, 1]

such that:

\int_{0}^{1} f (x) d x = 1 \int_{0}^{1} x f (x) d x = 1

Tags: functions, integration, calculus

Odd and even outputs

Let

g : N \times N \to N

be a function.

This means that

g

takes two natural number inputs and gives one natural number output. For example if

g

is defined by

g (n, m) = n + m

then

g (3, 4) = 7

and

g (10, 2) = 12

The function

g (n, m) = n + m

will give an even output if

n

and

m

are both odd or both even and an odd output if one is odd and the other is even. This could be summarised in the following table:

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	odd	even

Using only

+

and

\times

, can you construct functions

g (n, m)

which give the following output tables:

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	odd	odd

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	odd	even

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	even	odd

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	even	even

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	odd	odd

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	odd	even

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	even	odd

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	even	even

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	odd	odd

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	odd	even

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	even	odd

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	even	even

		$n$
		odd	even
$m$	odd	even	even
$m$	e	odd	odd

		$n$
		odd	even
$m$	odd	even	even
$m$	e	odd	even

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	odd

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	even

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		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	odd	odd

g (n, m) = 1

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	odd	even

g (n, m) = n \times m + n + m

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	even	odd

g (n, m) = n \times m + n + 1

		$n$
		odd	even
$m$	odd	odd	odd
$m$	e	even	even

g (n, m) = m

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	odd	odd

g (n, m) = n \times m + m + 1

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	odd	even

g (n, m) = n

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	even	odd

g (n, m) = n + m + 1

		$n$
		odd	even
$m$	odd	odd	even
$m$	e	even	even

g (n, m) = n \times m

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	odd	odd

g (n, m) = n \times m + 1

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	odd	even

g (n, m) = n + m

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	even	odd

g (n, m) = n + 1

		$n$
		odd	even
$m$	odd	even	odd
$m$	e	even	even

g (n, m) = n \times m + n

		$n$
		odd	even
$m$	odd	even	even
$m$	e	odd	odd

g (n, m) = m + 1

		$n$
		odd	even
$m$	odd	even	even
$m$	e	odd	even

g (n, m) = n \times m + n

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	odd

g (n, m) = n \times m + n + m + 1

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	even

g (n, m) = 2

Extension

Can you find functions

h : N \times N \times N \to N

(call the inputs

n

m

and

l

) to give the following outputs:

l

odd

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	even

l

even

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	even

l

odd

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	even

l

even

		$n$
		odd	even
$m$	odd	even	even
$m$	e	even	odd

etc

Tags: functions

Bézier curve

A Bézier curve is created as follows:

1) A set of points

P_{0}

, ...,

P_{n}

are chosen (in the example

n = 4

2) A set of points

Q_{0}

, ...,

Q_{n - 1}

are defined by

Q_{i} = t P_{i + 1} + (1 - t) P_{i}

(shown in green).

3) A set of points

R_{0}

, ...,

R_{n - 2}

are defined by

R_{i} = t Q_{i + 1} + (1 - t) Q_{i}

(shown in blue).

n

) After repeating the process

n

times, there will be one point. The Bézier curve is the path traced by this point at

t

varies between 0 and 1.

What is the Cartesian equation of the curve formed when:

P_{0} = (0, 1)

P_{1} = (0, 0)

P_{2} = (1, 0)

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P_{0} = (0, 1)

P_{1} = (0, 0)

P_{2} = (1, 0)

then

Q_{0} = t (0, 0) + (1 - t) (0, 1) = (0, 1 - t)

Q_{1} = t (1, 0) + (1 - t) (0, 0) = (t, 0)

and so

R_{0} = t (t, 0) + (1 - t) (0, 1 - t) = (t^{2}, (1 - t)^{2})

which means that

x = t^{2}

y = (1 - t)^{2}

y = (1 - \sqrt{x})^{2}

Extension

What should

P_{0}

P_{1}

and

P_{2}

be to get a curve with Cartesian equation

y = (1 - \sqrt{2 x})^{2}

Tags: functions, coordinates, graphs

End of puzzles.

Puzzles

14 December

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Find them all

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Extension

Odd and even outputs

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Extension

Bézier curve

Show answer & extension

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Extension

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