Dec 28 |
Semi-regular Tessellation on Hinges C |

Dec 26 |
Semi-regular Tessellation on Hinges B |

Dec 26 |
Semi-regular Tessellation on Hinges A |

Dec 24 |
A Hinged Realization of a Plane Tessellation |

Dec 23 |
A Corrected Version of the Calculus Proof of the Pythagorean Theorem |

Dec 23 |
Proof #89 of the Pythagorean Theorem |

Dec 22 |
Dancing Squares or a Hinged Plane Tessellation |

Dec 21 |
Filling a Grid with Good Neighbors |

Dec 20 |
Review of *A Decade of the Berkley Math Circle* |

Dec 20 |
No Flipping without End |

Dec 18 |
Viviani by Inversion |

Dec 17 |
Absolute Value and 1998 integers |

Dec 17 |
Word Ladders |

Dec 16 |
Phantom Circle and Recaptured Symmetry |

Dec 15 |
Eight Selections in Six Sectors |

Dec 14 |
A Game of Candy Squares |

Dec 13 |
Angles Inscribed in an Absent Circle |

Dec 12 |
A Three Pegs Question |

Dec 12 |
Escape of the Clones |

Dec 11 |
Guests at a Round Table |

Dec 11 |
Ptolemy by Inversion |

Dec 9 |
Treasure Hunt from Outside the Grid |

Dec 9 |
Treasure Hunt in a Square Grid |

Dec 7 |
12 Coins in Verse |

Nov 30 |
Menelaus from 3D |

Nov 30 |
Chromatic Number of the Plane |

Nov 29 |
The Rascal Triangle |

Nov 29 |
Coloring Points in the Plane |

Nov 28 |
Szilassi Polyhedron |

Nov 28 |
Csaszar Polyhedron 4 |

Nov 28 |
Csaszar Polyhedron 1 |

Nov 27 |
Acute Angle Bisectors in a Right Triangle |

Nov 27 |
1995 British Mathematical Olympiad, Problem 3 |

Nov 27 |
Circle through the Circumcenter |

Nov 26 |
Skyscrapers Puzzle |

Nov 25 |
Cevian Cradle |

Nov 24 |
Diophantine Equations |

Nov 24 |
Curious Identities Involving Integer Squares |

Nov 24 |
AM-GM Inequality |

Nov 23 |
VMO 1992 Problem 1 |

Nov 23 |
A Quickie Limit |

Nov 23 |
Any Integer with Three 2s |

Nov 21 |
Two-Parameter Solutions to Three Almost Fermat Equations |

Nov 21 |
A Review of *Geometric Etudes in Combinatorial Mathematics* |

Nov 20 |
Coloring Plane with Three Colors |

Nov 17 |
Tiling Rectangles with L-Trominoes |

Nov 15 |
Subtraction in a Rectangular Array |

Nov 13 |
Not Exceeding 24 |

Nov 13 |
Knots on a Torus |

Nov 12 |
Averaging Raindrops, Part 2 |

Nov 11 |
Averaging Raindrops - an exercise in geometric probability |

Nov 10 |
Solution #4 to a three circles problem |

Nov 7 |
100 Grasshoppers on a Triangular Board |

Oct 31 |
Polygon in a Rectangle |

Oct 31 |
Three Jugs - Equal Amounts |

Oct 30 |
Many Jugs to One II |

Oct 30 |
Many Jugs to One I |

Oct 30 |
3 Jugs Problem - A Water Doubling Variant |

Oct 29 |
Riemann Rearrangement Theorem |

Oct 28 |
1/2 = 1 |

Oct 27 |
Review of *An Introdcution to Diophantine Equations* by T. Andreescu *et al* |

Oct 27 |
A Short Equation in Reciprocals |

Oct 25 |
An Equation in Reciprocals |

Oct 25 |
Segment Trisection Induced by Parallels to Medians |

Oct 23 |
Angle Bisectors In Rectangle |

Oct 20 |
First Applications of Helly's Theorem |

Oct 20 |
Helly's Theorem |

Oct 19 |
Diophantine Quadratic Equation in Three Variables |

Oct 18 |
Dissection of Triangle into Rhombus by Hubert Shutrick |

Oct 18 |
Minus One But What a Difference |

Oct 14 |
Homothety between In- and Excircles |

Oct 13 |
Filling Pool with Fluids |

Oct 12 |
Calculus Proof of the Pythagorean Theorem |

Oct 11 |
Review of *Mathematical Plums* by R. Honsberger |

Oct 9 |
Dissection of Triangle into Rhombus |

Oct 8 |
Pentagon in a Semicircle |

Oct 7 |
Extremal Problem in a Circular Segment |

Oct 6 |
Counting Ordered Pairs |

Oct 4 |
Asymmetric Propeller by Plane Tiling |

Oct 2 |
A Mathematical Rabbit out of an Algebraic Hat |

Oct 1 |
Carpets in Hexagon |

Sep 29 |
Distinct Multisets with the Same Pairwise Sum |

Sep 28 |
A Characterization of the Euler Line |

Sep 27 |
Lennes' Polyhedron |

Sep 24 |
Review of *Classical Topics in Discrete Geometry* by K. Bezdek |

Sep 24 |
Morley's Theorem: Proof by H. Shutrick |

Sep 23 |
A Chain of Touching Circles in a Polygon (a la Bui Quang Tuan) |

Sep 23 |
A Chain of Touching Circles in a Polygon (a la Evelyn) |

Sep 21 |
Fraction Comparison Sped up |

Sep 19 |
Pythagoras' from the Star of David |

Sep 17 |
Areas in Square by Dissection |

Sep 14 |
Review of *Charming Proofs* by C. Alsina and R. B. Nelsen |

Sep 11 |
Antiparallel via Three Reflections |

Sep 10 |
Johnson Circles |

Sep 8 |
Assigning Numbers to Points in the Plane |

Sep 6 |
Squares in Semicircle and Circle |

Sep 5 |
Are Most Triangles Obtuse? |

Sep 3 |
Volumes of Two Pyramids |

Sep 3 |
Squares in Triangles |

Sep 1 |
Pythagorean Theorem: An Algebraic Proof with Three Trapezoids |

Aug 27 |
A Maximum Area Property of the Medial Triangle |

Aug 26 |
The Broken Chord Theorem: proof by Mariano Perez de la Cruz |

Aug 26 |
The Illustrated Law of Cosines |

Aug 16 |
Pythagoras' from Bottema's |

Aug 15 |
A Lemma of Equal Areas |

August 4 |
Touching Circles with Given Centers II |

August 4 |
Review of *Numbers Rule* by G. Szpiro |

August 3 |
2001 Poland Mathematical Olympiad, Problem 2 |

August 2 |
Are Australian Elections Fair As Claimed? |

August 2 |
Nested Radicals |

August 1 |
Pythagoras' from Broken Chords (proof #85) |

August 1 |
The Broken Chord Theorem: proof by Bui Quang Tuan |

August 1 |
Three Equal Circles in a Semicircle |

July 31 |
Shuffling Probability |

July 27 |
Two Triples of Concurrent Circles |

July 23 |
Six Circles Theorem (Bui Quang Tuan) |

July 23 |
Finger Counting |

July 16 |
Circle in a Square Inscribed in a Circle |

July 11 |
J. Casey's Additions to Euclid I.47 |

July 6 |
A Circle Rolling in an Equilateral Triangle |

June 27 |
Line, Circle, and Fixed Points |

June 26 |
An inequality from 1971 Canadian Mathematical Olympiad |

June 26 |
Problem with angle bisector from 1975 Canadian Mathematical Olympiad |

June 26 |
Fixed points from 2007 Irish Mathematical Olympiad |

June 25 |
Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad |

June 24 |
Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad |

June 23 |
Review of *What's Luck Has to Do with It?* by Joseph Mazur |

June 22 |
A Trigonometric Identity in a Right Triangle from the 1960 IMO |

June 22 |
Ancient Problem = Ancient Solution |

June 21 |
Euclid's Game on a Square Grid |

June 18 |
Square in a Circle Inscribed in a Square |

June 17 |
Parametric Problem in Quartic Polynomial from 1984 USAMO |

June 16 |
Example: A Poker Hand |

June 15 |
A Property of Centrally Symmetric Hexagons |

June 14 |
Chain of Circles on a Chord |

June 9 |
Circle, Isosceles Triangle and Fixed Point |

June 9 |
A Circle in an Isosceles triangle from the 2006 Irish MO |

June 8 |
Circles on the sides of a triangle from the 2005 Austrian MO |

June 8 |
Problem of divisibility from the 1970 Canadian MO |

June 8 |
Concurrency from the 2009 Austrian MO |

June 8 |
Problem in Isosceles Triangles from the 2008 Austrian MO |

June 7 |
Construction of Square from the 2008 Austrian MO |

June 7 |
A Problem with Ratios from the 1969 Canadian MO |

June 4 |
Cutting Triangles off Regular Hexagon |

June 4 |
Asian Pacific Kite |

June 3 |
Number Comparison - from the 1969 Canadian MO |

June 3 |
Lucky Contest Winners |

June 2 |
An Identity Involving the Integer Part - Problem from the 1981 Canadian MO |

June 2 |
An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO |

June 2 |
A Diophantine Equation with Square Roots - Problem from the 2008 British MO |

June 2 |
Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO |

June 2 |
Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO |

June 1 |
A Diophantine Equation - Problem from the 2000 Belarus MO |

May 30 |
Review of *A Mathematician's Lament* by Paul Lockhart |

May 28 |
Sangaku with Angle between a Tangent and a Chord |

May 27 |
Squares and Cubes - from the 1978 Canadian MO |

May 27 |
Locus of the midpoints of concurrent chords - from the 1991 Canadian MO |

May 27 |
Sum of 1984 consecutive squares - from the 1984 Canadian MO |

May 27 |
Areas in a quadrilateral - from the 1978 Canadian MO |

May 27 |
1986 Canadian MO, Problem 1 |

May 27 |
Circle through the Incenter And Antiparallels |

May 27 |
2001 IberoAmerican MO, Problem 2 |

May 27 |
1980 Canadian MO, Problem 3 |

May 27 |
1972 Canadian MO, Problem 5 |

May 27 |
1969 Canadian MO, Problem 7 |

May 27 |
1994 Ireland MO, Problem 1 |

May 27 |
Another Look at Nagel's Theorem |

May 26 |
1987 Canadian MO, Problem 2 |

May 21 |
Need Mathematics Taught in School Be Relevant? |

May 20 |
Yet another solution (by Vladimir Nikolin) to the "circle through 2 vertices and the incenter" problem |

May 19 |
Equilateral triangles on the sides of a right triangle - a problem from the 2000 Belarus MO |

May 19 |
Three Circles through the Incenter |

May 18 |
2000 Belarus Mathematical Olympiad, Problem 7 |

May 18 |
Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad |

May 18 |
Problem 7 of the Australian Mathematical Olympiad 2009 |

May 18 |
Problem 2 of the Australian Mathematical Olympiad 2008 |

May 18 |
Circle through the Incenter |

May 17 |
2010 Australian Mathematical Olympiad, Problem 6 |

May 17 |
Inequality Between the Perimeters of Two Triangles and the Distances Between Their Vertices |

May 16 |
Reflections of the circumcenter |

May 15 |
Partial But General Case Of Fermat's Last Theorem 2 |

May 14 |
Partial But General Case Of Fermat's Last Theorem |

May 14 |
Simple Somos Sequence - a Russian Olympiad problem |

May 14 |
Simple Somos Sequence Calculator |

May 12 |
Inequality with Logarithms |

May 12 |
Maximum Ratio of Similarity - Belarus Olympiad Problem |

May 10 |
Simple dissection of an acute triangle |

May 9 |
Three Orthogonal Circles Through Three Given Points |

May 6 |
A Variant Of the Pythagorean Theorem |

May 6 |
Triangle ABC is right iff sin^2(A) + sin^2(B) + sin^2(C) = 2 |

May 5 |
1992 Asian Pacific Mathematical Olympiad, Problem 2 |

May 5 |
1986 Vietnam Mathematical Olympiad, Problem 4 |

May 4 |
Parallelogram Iterations |

May 3 |
1962 Vietnam Mathematical Olympiad, Problem 4 |

May 3 |
2010 USA Olympiad, Problem 1 |

May 3 |
2010 USA Olympiad, Problem 3 |

May 3 |
2010 USA Olympiad, Problem 4 |

May 3 |
Nagel's Theorem: an Elementary Proof |

May 3 |
Presence of the Antipodes on a Circle |

May 3 |
Swapping Rows and Columns |

May 2 |
Two Equilateral Triangles |

May 1 |
Two Altitudes, One Midpoint |

April 29 |
Constructions Related To An Inaccessible Point |

April 29 |
A Telescoping Series |

April 29 |
Yet Another Proof of the Concurrency of the Altitudes |

April 27 |
Heron's Formula: a Proof |

April 26 |
Approximation Of Roots |

April 25 |
Nine Point Circle: an Elementary Proof |

April 25 |
Existence of the Euler Line: An Elementary Proof |

April 25 |
Problem 1 from the 1988 IberoAmerican Olympiad |

April 24 |
Reflections of the Orthocenter II |

April 23 |
Maximum Perimeter Property of the Incircle |

April 21 |
Problem 5 from the USA Mathematical Olympiad |

April 21 |
Addition and Subtraction of Integers |

April 19 |
Faulty Symmetry |

April 18 |
Triangulated Dinosaur |

April 17 |
Problem 9 from the 2009 Middle European Mathematical Olympiad |

April 16 |
Counting and Equivalent Fractions |

April 16 |
Scoring Misere: Two Heaps Perfect Strategy |

April 12 |
Problem 2 from the 1992 China Mathematical Olympiad |

April 12 |
Problem 3 from the 1992 IberoAmerican Mathematical Olympiad |

April 12 |
Problem 2 from the 2007 British Mathematical Olympiad |

April 9 |
Sine and Cosine of 15 Degrees Angle |

April 8 |
Equal Areas in Equilateral Triangle |

April 6 |
Angle Bisector in Equilateral Trapezoid |

April 6 |
Problem 1 of the 2010 China Mathematical Olympiad |

April 5 |
Two Lines - Two Circles |

April 4 |
What Is Line? |

April 4 |
What Is Point? |

April 1 |
Problem 7 of the Canadian Mathematical Olympiad 1971 |

March 31 |
Problem 2 of the Canadian Mathematical Olympiad 2010 |

March 31 |
Problem 3 of the Balkan Mathematical Olympiad 1988 |

March 30 |
Properties of GCD and LCM |

March 29 |
GCD and LCM by Plain Factorization |

March 26 |
Common Multiples and the Least Common Multiple |

March 25 |
Factors And Multiples |

March 24 |
IMO 1998 Problem 1 |

March 22 |
Cantor-Bernstein-Schroeder theorem, a Second Proof |

March 17 |
Addition and Subtraction Formulas for Sine and Cosine III |

March 17 |
Linear System with Parameter |

March 16 |
Listing All the Composite Numbers |

March 16 |
Lucky Numbers |

March 12 |
Factorial Products |

March 11 |
Function in the Plane That Vanishes |

March 10 |
The Size of a Class: Two Viewpoints |

March 9 |
Less than, Equal to, Greater Than Symbols |

March 9 |
Points Generated by the Nine Points |

March 8 |
Tangent Lines and Circles in Convex Quadrilateral |

March 7 |
The Ballot Lemma |

March 5 |
The Reflection Lemma |

March 4 |
Getting a Scout out of Desert |

March 3 |
Random Walks |

March 2 |
Van Schooten's Locus Problem |

March 2 |
Square out of a Quadrilateral |

Feb 27 |
Diagonal Argument |

Feb 25 |
A Trigonometric Solution to a Difficult Sangaku Problem |

Feb 24 |
A Possibly First Proof of the Concurrence of Altitudes |

Feb 24 |
Patterns in Pascal's Triangle |

Feb 23 |
Construction of Pascal's Triangle |

Feb 23 |
Short Construction of the Geometric Mean |

Feb 22 |
Treatise on Arithmetical Triangle |

Feb 19 |
A Fair Game of Chance |

Feb 19 |
Isosceles Tetrahedron |

Feb 13 |
Concyclic Points in Bride's Chair |

Feb 13 |
Joseph Keech in Bride's Chair |

Feb 12 |
New Proof of Euclid's Theorem |

Feb 12 |
Properties of the Figures in Euclid I.47 |

Feb 11 |
Four Crossing Circles |

Feb 10 |
Two Circles on Angle Bisector |

Feb 10 |
Coaxal Circles on Perpendicular Bisector |

Feb 6 |
Equal Chords in Crossing Circles |

Feb 6 |
Parallel Chords in Crossing Circles |

Feb 3 |
Monty Hall Problem Redux |

Jan 31 |
Monty Hall Dilemma's Simulation |

Jan 30 |
The Plain Butterfly Theorem |

Jan 29 |
Proof 18 of the Butterfly Theorem |

Jan 28 |
Unproperty of the Powers of 2 |

Jan 27 |
The Local-Global Principle |

Jan 25 |
Infinitude Of Primes From Legendre's Formula |

Jan 25 |
Legendre's Formula |

Jan 24 |
Half Turn, Reflection in Point |

Jan 23 |
Clubs in a Vector Space |

Jan 21 |
Triangulating Squares |

Jan 20 |
A Proof of the Coincidence of Altitudes by W. Chapple (1749) |

Jan 19 |
p-adic Expansions |

Jan 18 |
p-adic Numbers |

Jan 18 |
What Is Space Completion? |

Jan 15 |
Line through a Point in an Angle |

Jan 13 |
Mutually (Jointly) Independent Events |

Jan 12 |
Morley's Trisector Theorem: Proof by R. J. Webster |

Jan 12 |
Morley's Trisector Theorem: Proof by H. D. Grossman |

Jan 11 |
However You Solve It ... A Wonderful Equation |

Jan 11 |
Trigonometry by Paper Folding |

Jan 10 |
Isoperimetric Property of Equilateral Triangles |

Jan 10 |
Maximum Area Property of Equilateral Triangles |

Jan 9 |
Sangaku with Quadratic Optimization |

Jan 7 |
Equilateral Triangle on Parallel Lines |

Jan 5 |
Circles with Equal Collinear Chords |

Jan 3 |
Translated Triangles |

Jan 2 |
Find a Common Chord of Given Length |

Jan 1 |
Equilateral and 3-4-5 Triangles |