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How many solutions does the equation $|x| + 2|y| = 100$ have?

Can you prove the amazing relationship between the heights of the ladders?

Show that the alternating sign sum of squares produces triangle numbers.

Is there a secret hidden in the poetry?

Use the conjecture based on radicals to prove the Last Theorem of Fermat for $n \ge 6$.

Can you discover the radius of the ball at the bottom of the bag?

Where must the ball strike inside the square to return its point of origin?

What would be the diameter of a circular duct to feed 2, 3 or 4 circular cables?

Show that it is impossible not to form a triangle with all the edges the same colour

Prove that the segment joining the centres, $AB$, is a perpendicular bisector of the common chord $XY$.

Prove that circles determined by points on each side of a triangle and each vertex are concurrent.

Prove that there exists a sequence of $n$ consecutive composite numbers.

Prove that the product of four consecutive integers is always one less than a perfect square

What form must the constant take in the iterative formula for the limit to be integer?

Can you find the radius of the circle in the corner?

Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.

Given the three perimeters of a cuboid can you determine its volume?

Prove that 10

^{n}+1 is divisible by 11 iff`n`is oddFind the smallest number made up of the digits 1 through 9 which is divisible by 99.

Prove that there are infinitely many primitive solutions to the equation $x$

^{2}+ $y$^{2}= 2$z$^{2}.How many rectangles with integral length sides have an area equal in value to the perimeter?

With an equal chance of picking two discs the same colour, how many discs are in the bag?

Find the number of black discs in the game of chance

Prove that

`a`!`b`! =`a`! +`b`! + 2^{c}has a unique solutionCan you prove that (2

^{n})! is divisible by 2^{2n − 1}?Can you determine the depth of the well by timing how long it takes to hear the splash?

Prove that the ratio of adjacent terms in the Fibonacci sequence F

_{$n$+1}/F_{$n$}tends towards φ.Find the sum of a modified infinite Fibonacci series.

Prove that P(X = L) = P(X = L−1) for all positive integer values of L.

Prove that the sum of a proper fraction and its reciprocal can never be integer

What is the probability of winning the game of chance?

Prove that the minimum number of moves to completely reverse the positions of the coloured counters can never be square.

Prove that

`n`^{4}+ 3`n`^{2}+ 2 is never square.Find the conditions for when √

`a`−`b`= √`c`has a solution.Given a number has strictly increasing digits, what is the probability that it contains 5-digits?

Can you find the dimensions of an inscribed rectangle?

For which diagonal lengths is the area of the rectangle, which contains a unit square, integer?

Prove that log

_{$a$}($x$)log_{$b$}($y$) = log_{$b$}($x$)log_{$a$}($y$).Can you find the missing length of the trapzium?

Can you find the shaded area of the lunes?

Is the claim about the average number of matches statistically significant?

Prove that (

`a`+`b`)/2 ≥ √(`ab`)Investigate the nature of the second order recurrence relation.

Prove that the difference between the expressions $x$

^{3}+ $y$ and $x$ + $y$^{3}is a multiple of six.Determine the nature of all multiplicatively perfect numbers.

Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value.

Prove that $6^n + 8^n$ is divisible by 7 iff $n$ is odd.

Prove that in any graph there will always be an even number of odd vertices

How many differen ways can you form five primes using the digits 0 to 9?

Prove that a quadrilateral has opposite sides of equal length if and only if it is a parallelogram

Investigating the number of paving a pathway.

How many rectangles exist for which the number of tiles on the perimeter are equal to the number of tiles on the inside?

Can you determine the fraction of the regular pentagon that the star occupies?

Prove that all even perfect numbers are triangle numbers.

Prove that the constructed triangle inside the unit square is isosceles but not equilateral

Prove that there can be no more than five regular (convex) polyhedra

Prove that the roots of the polynomial, $x$

^{n}+ $c$_{n-1}$x$^{n-1}+ ... + $c$_{2}$x$^{2}+ $c$_{1}$x$ + $c$_{0}= 0, are irrational or integer.Can you prove that the expression 21

^{n}− 5^{n}+ 8^{n}is divisible by 24?Find the least value of $x$ for which x

^{x}+ 1 is divisible by 2^{n}.Prove that 8

^{n}−1 is always divisible by 7Prove that there exists no prime which is one less than a multiple of four that can be written as the sum of two squares.

Given that $a$

^{2}+ $b$ and $a$ + $b$^{2}differ by prime amount find $a$ and $b$.Given that $p$ is prime, when is $4^p + p^4$ prime?

Given that $p$ is prime, when is $2^p + p^2$ prime?

Prove that if $a^n - 1$ is prime and $n \ge 2$ then $a = 2$ and $n$ must be prime.

Prove that there only ever exists one prime value $p$ for which $p + 1$ is a perfect power of $k$ and determine the condition for this perfect power to exist.

Prove that 2

^{p}+ 3^{p}can never be a perfect square.Prove that the for every primitive Pythagorean triplet ($a$, $b$, $c$) that $abc$ is divisible by sixty.

Show that the midpoints of any quadrilateral form a parallelogram

Prove that the radical axis of two intersecting circles is their common secant.

When does the quadratic,

`ax`^{2}−`px`−`p`= 0, have rational roots?Prove the a quadratic with non-zero integral coefficients can be found for which every arrangement of coefficients yields rational roots.

Solve the Diophantine equation

`c`=`a`/`b`+`b`/`a`Prove that

`p`and`p`^{2}cannot be permutations of the value of their Totient functionProve that the sum of two 2-digit numbers, for which the reverse of their digits forms a different pair of 2-digit numbers with the same sum, is always divisible by 11.

Can you show that every term in the sequence is divisibly by 18?

Find the fraction of the square which the shaded octagon fills.

Find the radius of the shaded circle.

Find the shortest distance from one corner of a cuboid to the opposite corner.

When is 8

`p`+1 square?Prove that sum of the squares of $x$ and $y$ can never be a multiple of their product.

Prove that N is a sum-product number iff it is composite.

Prove that the sum of the first $n$ terms in the Fibonacci sequence is given by F

_{n+2}− 1.Prove that three touching circles with a common tangent hold a special relationship between their radii

Find the probability that the triangle formed by the hands on a clock is isosceles

How many trailing zeroes does 1000! contain?

Find the maximum area of the triangle region in the unit square

Prove that each median in a triangle is split in the ratio 2:1.

Find the series of the reciprocals of triangle numbers

When is 8

`p`+1 a triangle number?Given that $n$ is a positive integer and 2$n$ + 1 and 3$n$ + 1 are perfect squares, prove that $n$ is divisible by 40.

Evaluate the exact value of the series.

Given that $x$

^{2}+ $x$ + 1 = 0, find the value of $x$^{3}.Show that a unique circle must pass through the vertices of the two triangles

Are you able to complete the email challenge?

How many 5-digit ZIP codes are detour-prone?