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15 Degree Triangle      Problem ID: 361 (11 Oct 2009)
Prove that the two smaller angles in the triangle are exactly 15o and 75o respectively.
Algebraic Cosine      Problem ID: 369 (30 Nov 2009)
Prove that cos($x$) is algebraic if $x$ is a rational multiple of Pi.
Almost Equilateral Triangles      Problem ID: 219 (30 Mar 2005)
Prove that infintitely many almost equilateral triangles exist
Composite Fibonacci Terms      Problem ID: 365 (03 Nov 2009)
Prove that for all composite values of $n$ > 4, Fn is composite.
Concurrent Congruent Segments      Problem ID: 309 (04 Feb 2007)
For the given construction prove that the segments are congruent and concurrent with one another.
Concurrent Segments In A Triangle      Problem ID: 316 (18 Mar 2007)
Prove that the product of given ratios in any triangle is always one.
Contains The Origin      Problem ID: 237 (02 Aug 2005)
Which shape is more likely to contain the origin?
Continued Fraction Irreducible Convergents      Problem ID: 286 (01 Aug 2006)
Prove that the convergents generated by the recurrence relation for continued fractions are irreducible.
Continued Fraction Recurrence Relation      Problem ID: 282 (15 Jul 2006)
Prove that the recurrence relation holds for continued fractions.
Counting Digits      Problem ID: 128 (Oct 2003)
How many digits does the number 21000 contain?
Diophantine Challenge      Problem ID: 227 (04 Jun 2005)
Solve the Diophantine equation, x2 + (bx)y = ±1
Euler Rules      Problem ID: 242 (16 Oct 2005)
Prove that Euler's rule, V + R = E + 2, is true for all planar graphs
Even Perfect Numbers      Problem ID: 326 (26 Jun 2007)
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.
Even Sum Of Two Abundant Numbers      Problem ID: 347 (08 Nov 2008)
Prove that every even $n$ ≥ 48 can be written as the sum of two abundant numbers.
Every Primitive Triplet      Problem ID: 302 (02 Jan 2007)
Prove that the given identities will produce every primitive Pythagorean triplet.
Exponential Symmetry      Problem ID: 344 (09 Jul 2008)
Given that $m$ and $n$ are positive integers, solve $m$n = $n$m.
Factorial And Square      Problem ID: 220 (30 Mar 2005)
Solve the equation a!b! = a! + b! + c2
Factorial Equation      Problem ID: 216 (09 Mar 2005)
Solve the factorial equation a!b! = a! + b! + c!
Farey Sequence      Problem ID: 181 (Oct 2004)
What fraction is the immediate predecessor of 2/5 in F100 of the Farey sequence?
Fibonacci Sequence      Problem ID: 135 (Nov 2003)
Prove that the given nth term formula for the Fibonacci sequence is true
Finishing With 99      Problem ID: 141 (Dec 2003)
Prove that 99n ends in 99 for odd n
Firework Rocket      Problem ID: 161 (Mar 2004)
How long does the rocket take to reach its maximum height?
Fourth Power Plus Four Prime      Problem ID: 355 (23 Jul 2009)
Given that $n$ is a positive integer, when is $n$4 + 4 prime?
General Factorial      Problem ID: 251 (02 Dec 2005)
Show that the Gamma function is a suitable extension of the factorial function
Geometric Division      Problem ID: 304 (12 Jan 2007)
Prove that the quotient and remainder for any prime cannot be in a geometric sequence.
Harmonic Sum Approximation      Problem ID: 209 (17 Feb 2005)
Estimate the sum of the first one hundred Harmonic numbers
Impossible Quadratic      Problem ID: 244 (19 Oct 2005)
Show that px2qx + q = 0 has no rational solutions
Infinite Circles      Problem ID: 367 (15 Nov 2009)
What fraction of the large red circle do the infinite set of smaller circles represent?
Integer Integral      Problem ID: 148 (Jan 2004)
Integrate the integer part function of 10$x$
Irrationality Of E      Problem ID: 377 (17 Oct 2010)
Prove that $e$ is irrational.
Irrationality Of Pi      Problem ID: 371 (24 Dec 2009)
Prove that π is irrational.
Irrational Cosine      Problem ID: 280 (13 May 2006)
Prove that $\cos(1^o)$ is irrational.
Loan Repayments      Problem ID: 260 (09 Jan 2006)
Find the fixed monthly payments to repay a loan of £2000 in thirty-six months at an annual rate of 12%
Lucky Dip      Problem ID: 276 (21 Apr 2006)
Find the expected return on this game of chance.
Maximised Box      Problem ID: 121 (May 2003)
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.
Maximum Product      Problem ID: 334 (19 Nov 2007)
Investigate sets for which the sum of elements is fixed and the product is maximised.
Napoleon Triangle      Problem ID: 313 (15 Feb 2007)
Prove that the centres joining the constructed equilateral triangles from any triangle always form an equilateral triangle.
Never Decreasing Digits      Problem ID: 263 (29 Jan 2006)
How many numbers below one million have increasing digits?
Never Prime      Problem ID: 235 (31 Jul 2005)
Prove that 14n + 11 is never prime
Odd Perfect Numbers      Problem ID: 328 (05 Jul 2007)
Prove that an odd perfect number must be of the form $c$2 $q$4$k$+1 where $q$ ≡ 1 mod 4 is prime.
Order Of A Prime      Problem ID: 288 (09 Sep 2006)
Prove that $p - 1$ is always a multiple of $ord(a, p)$.
Pairwise Products      Problem ID: 224 (24 May 2005)
Prove that the sum of pairwise products for a set of three real numbers whose sum is one cannot exceed one third
Perfect Digit      Problem ID: 331 (29 Aug 2007)
Prove that the last digit of an even perfect number will be 6 or 8.
Perfect Power Sum      Problem ID: 307 (20 Jan 2007)
Show that $x$$n$ + $y$$n$ = $p$ has no solution if $n$ contains an odd factor greater than one.
Perpendicular Medians      Problem ID: 296 (10 Dec 2006)
Investigating the properties of a triangle for which two of its medians are perpendicular.
Prime Partner      Problem ID: 291 (22 Sep 2006)
Prove that for every odd prime, $p$, there exists a unique positive integer, $n$, such that $n^2 + np$ is a perfect square.
Prime Reciprocals      Problem ID: 238 (02 Aug 2005)
Prove that there exist no set of primes for which the sum of reciprocals is integer
Primitive Pythagorean Triplets      Problem ID: 205 (24 Jan 2005)
Prove that the given matrix transformation will transform a primitive Pythagorean triplet into a new one.
Quadratic Circle      Problem ID: 168 (Apr 2004)
Find the co-ordinate of the point on the y-axis generated by the quadratic and the circle
Quadratic Differences      Problem ID: 295 (26 Nov 2006)
Determining the values of $n$ for which the quadratic equation has no solutions.
Quarter Circles      Problem ID: 142 (Dec 2003)
Find the area of the shaded region generated by four overlapping quarter circles
Radical Convergence      Problem ID: 257 (01 Jan 2006)
Prove that the Pell equation $x^2 - dy^2 = 1$ can be used to find convergents for $\sqrt{d}$
Random Chords      Problem ID: 195 (21 Dec 2004)
Determine the probability that all r random chords are non-intersecting.
Reciprocal Radical Sum      Problem ID: 266 (05 Feb 2006)
When is the sum of reciprocal square roots rational?
Reciprocal Sum      Problem ID: 169 (Apr 2004)
For a given x, determine the number of solutions of 1/x+1/y=1/z
Repunit Divisibility      Problem ID: 293 (03 Oct 2006)
If $GCD(n, 10) = 1$, prove that there exists some repunit which is divisible by $n$.
Sliding Box      Problem ID: 155 (Feb 2004)
Can you find the coefficient of friction between the two surfaces?
Square Lattice Triangles      Problem ID: 162 (Mar 2004)
Find the probability that three randomly chosen points of a 5 by 5 lattice will form a triangle
Square Sum Is Square      Problem ID: 336 (27 Mar 2008)
Prove that there exists a sum of $n$ distinct squares that is also square.
String Of Ones And Zeroes      Problem ID: 187 (01 Nov 2004)
Prove that there always exists a number made up of ones and zeroes that is divisible by the positive integer, n
Sum Of Two Abundant Numbers      Problem ID: 348 (08 Nov 2008)
Prove that all integers greater than 28123 can be written as the sum of two abundant numbers.
Tangent Sum And Product      Problem ID: 115 (Apr 2003)
Can you prove that the sum of the tangents is equal to the product?
Time Loses Integrity      Problem ID: 272 (16 Feb 2006)
Prove that the time the body takes to pass between two point can never be integer.
Unique Square Sum      Problem ID: 176 (May 2004)
Prove that if a prime is expressible as the sum of two squares it can be done in only one way