mathschallenge.net

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 21^$n$ − 5^$n$ + 8^$n$ is divisible by 24?

Powerful Divisor

Problem ID: 319 (07 Apr 2007)

Find the least value of $x$ for which x^x + 1 is divisible by 2ⁿ.

Power Divisibility

Problem ID: 204 (24 Jan 2005)

Prove that 8^$n$−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 2^$p$ + 3^$p$ 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, $ax$² − $px$ − $p$ = 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 $p$² 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 8$p$+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 F_n+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 8$p$+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?

15 Degree Triangle

Problem ID: 361 (11 Oct 2009)

Prove that the two smaller angles in the triangle are exactly 15^o and 75^o 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, F_n 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, $x$² + ($b$−$x$)$y$ = ±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 $a$!$b$! = $a$! + $b$! + $c$²

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 $n$^th term formula for the Fibonacci sequence is true

Finishing With 99

Problem ID: 141 (Dec 2003)

Prove that 99^$n$ 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 $px$² − $qx$ + $q$ = 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 14^$n$ + 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