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Absolute Sum      Problem ID: 292 (03 Oct 2006)
How many solutions does the equation $|x| + 2|y| = 100$ have?
Alley Ladders      Problem ID: 127 (Oct 2003)
Can you prove the amazing relationship between the heights of the ladders?
Alternating Sign Square Sum      Problem ID: 262 (29 Jan 2006)
Show that the alternating sign sum of squares produces triangle numbers.
Anonymous Author      Problem ID: 3 (Aug 2000)
Is there a secret hidden in the poetry?
A Radical Proof      Problem ID: 284 (23 Jul 2006)
Use the conjecture based on radicals to prove the Last Theorem of Fermat for $n \ge 6$.
Bag Of Balls      Problem ID: 139 (Dec 2003)
Can you discover the radius of the ball at the bottom of the bag?
Bouncing Ball      Problem ID: 363 (28 Oct 2009)
Where must the ball strike inside the square to return its point of origin?
Cables      Problem ID: 28 (Dec 2000)
What would be the diameter of a circular duct to feed 2, 3 or 4 circular cables?
Coloured Strings      Problem ID: 259 (09 Jan 2006)
Show that it is impossible not to form a triangle with all the edges the same colour
Common Chord      Problem ID: 312 (15 Feb 2007)
Prove that the segment joining the centres, $AB$, is a perpendicular bisector of the common chord $XY$.
Concurrent Circles In A Triangle      Problem ID: 321 (14 Apr 2007)
Prove that circles determined by points on each side of a triangle and each vertex are concurrent.
Consecutive Composites      Problem ID: 325 (26 Jun 2007)
Prove that there exists a sequence of $n$ consecutive composite numbers.
Consecutive Product Square      Problem ID: 184 (01 Nov 2004)
Prove that the product of four consecutive integers is always one less than a perfect square
Converging Root      Problem ID: 160 (Mar 2004)
What form must the constant take in the iterative formula for the limit to be integer?
Corner Circle      Problem ID: 84 (Oct 2002)
Can you find the radius of the circle in the corner?
Cubes And Multiples Of 7      Problem ID: 24 (Nov 2000)
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.
Cuboid Perimeters To Volume      Problem ID: 357 (26 Aug 2009)
Given the three perimeters of a cuboid can you determine its volume?
Divisible By 11      Problem ID: 208 (17 Feb 2005)
Prove that 10n+1 is divisible by 11 iff n is odd
Divisible By 99      Problem ID: 359 (27 Sep 2009)
Find the smallest number made up of the digits 1 through 9 which is divisible by 99.
Double Square Sum      Problem ID: 333 (19 Nov 2007)
Prove that there are infinitely many primitive solutions to the equation $x$2 + $y$2 = 2$z$2.
Equable Rectangles      Problem ID: 340 (18 Jun 2008)
How many rectangles with integral length sides have an area equal in value to the perimeter?
Equal Chance      Problem ID: 146 (Jan 2004)
With an equal chance of picking two discs the same colour, how many discs are in the bag?
Equal Colours      Problem ID: 193 (11 Dec 2004)
Find the number of black discs in the game of chance
Factorial And Power Of 2      Problem ID: 215 (09 Mar 2005)
Prove that a!b! = a! + b! + 2c has a unique solution
Factorial Divisibility      Problem ID: 40 (Mar 2001)
Can you prove that (2n)! is divisible by 22n − 1?
Falling Sound      Problem ID: 346 (21 Sep 2008)
Can you determine the depth of the well by timing how long it takes to hear the splash?
Fibonacci Ratio      Problem ID: 311 (15 Feb 2007)
Prove that the ratio of adjacent terms in the Fibonacci sequence F$n$+1/F$n$ tends towards φ.
Fibonacci Series      Problem ID: 299 (17 Dec 2006)
Find the sum of a modified infinite Fibonacci series.
Fishy Problem      Problem ID: 324 (30 May 2007)
Prove that P(X = L) = P(X = L−1) for all positive integer values of L.
Fraction Reciprocal Sum      Problem ID: 153 (Feb 2004)
Prove that the sum of a proper fraction and its reciprocal can never be integer
Highest Roll Wins      Problem ID: 158 (Mar 2004)
What is the probability of winning the game of chance?
Hops And Slides But Never Square      Problem ID: 372 (07 Aug 2010)
Prove that the minimum number of moves to completely reverse the positions of the coloured counters can never be square.
Imperfect Square Sum      Problem ID: 133 (Nov 2003)
Prove that n4 + 3n2 + 2 is never square.
Impossible Solution      Problem ID: 190 (28 Nov 2004)
Find the conditions for when √ab = √c has a solution.
Increasing Digits      Problem ID: 180 (Oct 2004)
Given a number has strictly increasing digits, what is the probability that it contains 5-digits?
Inscribed Rectangle      Problem ID: 63 (Jan 2002)
Can you find the dimensions of an inscribed rectangle?
Integral Area      Problem ID: 223 (24 May 2005)
For which diagonal lengths is the area of the rectangle, which contains a unit square, integer?
Inverted Logarithm      Problem ID: 315 (18 Mar 2007)
Prove that log$a$($x$)log$b$($y$) = log$b$($x$)log$a$($y$).
Isosceles Trapezium      Problem ID: 79 (May 2002)
Can you find the missing length of the trapzium?
Lunes      Problem ID: 20 (Oct 2000)
Can you find the shaded area of the lunes?
Mean Claim      Problem ID: 306 (20 Jan 2007)
Is the claim about the average number of matches statistically significant?
Mean Proof      Problem ID: 230 (10 Jul 2005)
Prove that (a + b)/2 ≥ √(ab)
Mean Sequence      Problem ID: 212 (06 Mar 2005)
Investigate the nature of the second order recurrence relation.
Multiple Of Six Difference      Problem ID: 330 (13 Jul 2007)
Prove that the difference between the expressions $x$3 + $y$ and $x$ + $y$3 is a multiple of six.
Multiplicatively Perfect      Problem ID: 342 (26 Jun 2008)
Determine the nature of all multiplicatively perfect numbers.
Multiplying Magic Square      Problem ID: 374 (16 Aug 2010)
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.
Odd Power Divisibility      Problem ID: 279 (13 May 2006)
Prove that $6^n + 8^n$ is divisible by 7 iff $n$ is odd.
Odd Vertices      Problem ID: 254 (12 Dec 2005)
Prove that in any graph there will always be an even number of odd vertices
Pandigital Primes      Problem ID: 185 (01 Nov 2004)
How many differen ways can you form five primes using the digits 0 to 9?
Parallelogram Property      Problem ID: 245 (28 Oct 2005)
Prove that a quadrilateral has opposite sides of equal length if and only if it is a parallelogram
Pathway Arrangements      Problem ID: 270 (16 Feb 2006)
Investigating the number of paving a pathway.
Peculiar Perimeter      Problem ID: 89 (Nov 2002)
How many rectangles exist for which the number of tiles on the perimeter are equal to the number of tiles on the inside?
Pentagon Star      Problem ID: 271 (16 Feb 2006)
Can you determine the fraction of the regular pentagon that the star occupies?
Perfect Triangles      Problem ID: 329 (13 Jul 2007)
Prove that all even perfect numbers are triangle numbers.
Perpendicular Construction      Problem ID: 134 (Nov 2003)
Prove that the constructed triangle inside the unit square is isosceles but not equilateral
Platonic Solids      Problem ID: 246 (28 Oct 2005)
Prove that there can be no more than five regular (convex) polyhedra
Polynomial Roots      Problem ID: 373 (07 Aug 2010)
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.
Powerful Divisibility      Problem ID: 32 (Jan 2001)
Can you prove that the expression 21n − 5n + 8n is divisible by 24?
Powerful Divisor      Problem ID: 319 (07 Apr 2007)
Find the least value of $x$ for which xx + 1 is divisible by 2n.
Power Divisibility      Problem ID: 204 (24 Jan 2005)
Prove that 8n−1 is always divisible by 7
Primes And Square Sums      Problem ID: 120 (May 2003)
Prove that there exists no prime which is one less than a multiple of four that can be written as the sum of two squares.
Prime Difference      Problem ID: 322 (14 Apr 2007)
Given that $a$2 + $b$ and $a$ + $b$2 differ by prime amount find $a$ and $b$.
Prime Exponent And Fourth Power Sum      Problem ID: 269 (11 Feb 2006)
Given that $p$ is prime, when is $4^p + p^4$ prime?
Prime Exponent And Square Sum      Problem ID: 265 (05 Feb 2006)
Given that $p$ is prime, when is $2^p + p^2$ prime?
Prime Form      Problem ID: 285 (01 Aug 2006)
Prove that if $a^n - 1$ is prime and $n \ge 2$ then $a = 2$ and $n$ must be prime.
Prime Power      Problem ID: 290 (22 Sep 2006)
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.
Prime Power Sum      Problem ID: 191 (28 Nov 2004)
Prove that 2p + 3p can never be a perfect square.
Pythagorean Triplet Product      Problem ID: 301 (02 Jan 2007)
Prove that the for every primitive Pythagorean triplet ($a$, $b$, $c$) that $abc$ is divisible by sixty.
Quadrilateral Parallelogram      Problem ID: 250 (02 Dec 2005)
Show that the midpoints of any quadrilateral form a parallelogram
Radical Axis      Problem ID: 194 (21 Dec 2004)
Prove that the radical axis of two intersecting circles is their common secant.
Rational Quadratic      Problem ID: 247 (28 Oct 2005)
When does the quadratic, ax2pxp = 0, have rational roots?
Rational Roots Quadratic      Problem ID: 274 (21 Apr 2006)
Prove the a quadratic with non-zero integral coefficients can be found for which every arrangement of coefficients yields rational roots.
Reciprocal Symmetry      Problem ID: 241 (16 Oct 2005)
Solve the Diophantine equation c = a/b + b/a
Relatively Prime Permutations      Problem ID: 240 (10 Aug 2005)
Prove that p and p2 cannot be permutations of the value of their Totient function
Reverse Equivalence      Problem ID: 126 (Oct 2003)
Prove 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.
Sequence Divisibility      Problem ID: 75 (Apr 2002)
Can you show that every term in the sequence is divisibly by 18?
Shaded Octagon      Problem ID: 298 (17 Dec 2006)
Find the fraction of the square which the shaded octagon fills.
Sixteen Circles      Problem ID: 186 (01 Nov 2004)
Find the radius of the shaded circle.
Spider Fly Distance      Problem ID: 201 (10 Jan 2005)
Find the shortest distance from one corner of a cuboid to the opposite corner.
Square Search      Problem ID: 231 (10 Jul 2005)
When is 8p+1 square?
Sum Of Squares And Multiple Of Product      Problem ID: 300 (02 Jan 2007)
Prove that sum of the squares of $x$ and $y$ can never be a multiple of their product.
Sum Product Numbers      Problem ID: 200 (10 Jan 2005)
Prove that N is a sum-product number iff it is composite.
The Fibonacci Series      Problem ID: 352 (17 Apr 2009)
Prove that the sum of the first $n$ terms in the Fibonacci sequence is given by Fn+2 − 1.
Three Circles      Problem ID: 175 (May 2004)
Prove that three touching circles with a common tangent hold a special relationship between their radii
Tick Tock Triangle      Problem ID: 167 (Apr 2004)
Find the probability that the triangle formed by the hands on a clock is isosceles
Trailing Zeroes      Problem ID: 174 (May 2004)
How many trailing zeroes does 1000! contain?
Triangle In Square      Problem ID: 147 (Jan 2004)
Find the maximum area of the triangle region in the unit square
Triangle Median      Problem ID: 213 (06 Mar 2005)
Prove that each median in a triangle is split in the ratio 2:1.
Triangle Reciprocals      Problem ID: 236 (02 Aug 2005)
Find the series of the reciprocals of triangle numbers
Triangle Search      Problem ID: 234 (31 Jul 2005)
When is 8p+1 a triangle number?
Two Squares      Problem ID: 338 (16 May 2008)
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.
Unexpected Sum      Problem ID: 275 (21 Apr 2006)
Evaluate the exact value of the series.
Uninvited Guest      Problem ID: 354 (23 Jul 2009)
Given that $x$2 + $x$ + 1 = 0, find the value of $x$3.
Unique Circle Equal Angles      Problem ID: 249 (27 Nov 2005)
Show that a unique circle must pass through the vertices of the two triangles
XOR Challenge      Problem ID: 1 (Aug 2000)
Are you able to complete the email challenge?
Zip Codes      Problem ID: 154 (Feb 2004)
How many 5-digit ZIP codes are detour-prone?