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Prove that the two smaller angles in the triangle are exactly 15

^{o}and 75^{o}respectively.Prove that cos($x$) is algebraic if $x$ is a rational multiple of Pi.

Prove that infintitely many

`almost equilateral triangles`existProve that for all composite values of $n$ > 4, F

_{n}is composite.For the given construction prove that the segments are congruent and concurrent with one another.

Prove that the product of given ratios in any triangle is always one.

Which shape is more likely to contain the origin?

Prove that the convergents generated by the recurrence relation for continued fractions are irreducible.

Prove that the recurrence relation holds for continued fractions.

How many digits does the number 2

^{1000}contain?Solve the Diophantine equation,

`x`^{2}+ (`b`−`x`)`y`= ±1Prove that Euler's rule, V + R = E + 2, is true for all planar graphs

Prove that P is an even perfect number iff it is of the form $2^{n-1}(2^n - 1)$ where $2^n - 1$ is prime.

Prove that every even $n$ ≥ 48 can be written as the sum of two abundant numbers.

Prove that the given identities will produce every primitive Pythagorean triplet.

Given that $m$ and $n$ are positive integers, solve $m$

^{n}= $n$^{m}.Solve the equation

`a`!`b`! =`a`! +`b`! +`c`^{2}Solve the factorial equation

`a`!`b`! =`a`! +`b`! +`c`!What fraction is the immediate predecessor of 2/5 in F

_{100}of the Farey sequence?Prove that the given

`n`^{th}term formula for the Fibonacci sequence is trueProve that 99

^{n}ends in 99 for odd`n`How long does the rocket take to reach its maximum height?

Given that $n$ is a positive integer, when is $n$

^{4}+ 4 prime?Show that the Gamma function is a suitable extension of the factorial function

Prove that the quotient and remainder for any prime cannot be in a geometric sequence.

Estimate the sum of the first one hundred Harmonic numbers

Show that

`px`^{2}−`qx`+`q`= 0 has no rational solutionsWhat fraction of the large red circle do the infinite set of smaller circles represent?

Integrate the integer part function of 10

^{x}Prove that $e$ is irrational.

Prove that π is irrational.

Prove that $\cos(1^o)$ is irrational.

Find the fixed monthly payments to repay a loan of £2000 in thirty-six months at an annual rate of 12%

Find the expected return on this game of chance.

Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.

Investigate sets for which the sum of elements is fixed and the product is maximised.

Prove that the centres joining the constructed equilateral triangles from any triangle always form an equilateral triangle.

How many numbers below one million have increasing digits?

Prove that 14

^{n}+ 11 is never primeProve that an odd perfect number must be of the form $c$

^{2}$q$^{4$k$+1}where $q$ ≡ 1 mod 4 is prime.Prove that $p - 1$ is always a multiple of $ord(a, p)$.

Prove that the sum of pairwise products for a set of three real numbers whose sum is one cannot exceed one third

Prove that the last digit of an even perfect number will be 6 or 8.

Show that $x$

^{$n$}+ $y$^{$n$}= $p$ has no solution if $n$ contains an odd factor greater than one.Investigating the properties of a triangle for which two of its medians are perpendicular.

Prove that for every odd prime, $p$, there exists a unique positive integer, $n$, such that $n^2 + np$ is a perfect square.

Prove that there exist no set of primes for which the sum of reciprocals is integer

Prove that the given matrix transformation will transform a primitive Pythagorean triplet into a new one.

Find the co-ordinate of the point on the y-axis generated by the quadratic and the circle

Determining the values of $n$ for which the quadratic equation has no solutions.

Find the area of the shaded region generated by four overlapping quarter circles

Prove that the Pell equation $x^2 - dy^2 = 1$ can be used to find convergents for $\sqrt{d}$

Determine the probability that all

`r`random chords are non-intersecting.When is the sum of reciprocal square roots rational?

For a given

`x`, determine the number of solutions of 1/`x`+1/`y`=1/`z`If $GCD(n, 10) = 1$, prove that there exists some repunit which is divisible by $n$.

Can you find the coefficient of friction between the two surfaces?

Find the probability that three randomly chosen points of a 5 by 5 lattice will form a triangle

Prove that there exists a sum of $n$ distinct squares that is also square.

Prove that there always exists a number made up of ones and zeroes that is divisible by the positive integer,

`n`Prove that all integers greater than 28123 can be written as the sum of two abundant numbers.

Can you prove that the sum of the tangents is equal to the product?

Prove that the time the body takes to pass between two point can never be integer.

Prove that if a prime is expressible as the sum of two squares it can be done in only one way