Geometry puzzles
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We will post the names of anyone who solves any of these puzzles. The harder puzzles are marked * and **. If the answer is not known, then the puzzle will be marked ##.



The Folding Ruler
       I own one of those folding rulers where each segment is exactly 1 foot long. While playing with the open ruler I formed it into a triangle. Then I opened and refolded it into a second triangle with double the area. What is the smallest possible length of the ruler? What if the second triangle has 3 times the area?

Solved by:   Lee Morgenstern, Nick McGrath, P.M.A. Hakeem, Paul Cleary


The Folding Ruler #2
       I own one of those folding rulers where each segment is exactly 1 foot long. While playing with the open ruler I formed it into a triangle. Then I opened and refolded it into a second triangle with double the area. Next I opened and refolded it into a third triangle with triple the area. What is the smallest possible length of the ruler?

Solved by:   Nick McGrath, Lee Morgenstern, P.M.A. Hakeem, Paul Cleary


Triangle Ratios
       For what rational numbers a/b is it possible to find two triangles with integer sides whose areas are in the ratio a:b?

Solved by:   Lee Morgenstern


Bowtie
       Two line segments AB and CD intersect at E. Is it possible for the lengths of the 6 line segments AD, AE, BC, BE, CE and DE to be consecutive integers (in any order)?

Solved by:   Lee Morgenstern


Pie
       It is easy to cut a circular pie into 7 pieces using 3 straight cuts. How should the cuts be made so that the 7 pieces are as even as possible? (That is, the ratio between the largest and smallest piece should be minimal.)

Solved by:   Fotos Fotiadis


Make A Square
       What is the smallest number of pieces that can be assembled into a square in 2 distinct ways? 3 distinct ways? Solutions where the pieces, or the square, are merely rotated or reflected (turned over) will not be considered distinct.

Solved by:   Lee Morgenstern


Pythagorean Triangles #1A
       Find two Pythagorean right triangles A,B,C and D,E,F such that A+D is a square, B+E is a square, and C+F is a square. If this is impossible, find the nearest miss.

Solved by:   Lee Morgenstern, Paul Cleary, P.M.A. Hakeem


Pythagorean Triangles #1B
       Find two Pythagorean right triangles A,B,C and D,E,F such that A+D is a cube, B+E is a cube, and C+F is a cube. If this is impossible, find the nearest miss.

Solved by:   Lee Morgenstern


Pythagorean Triangles #1C
       Find two Pythagorean right triangles A,B,C and D,E,F such that A+D is a cube, B+E is a sixth power, and C+F is a ninth power. If this is impossible, find the nearest miss.

Solved by:   Lee Morgenstern


Pythagorean Triangles #2 (contributed by Lee Morgenstern)
       Find two Pythagorean right triangles A,B,C and D,E,F such that A=D and B=F. If that is impossible, then find a pair such that A=D and abs(B-F) is minimum.

Solved by:   Paul Cleary, P.M.A. Hakeem


Pythagorean Triangles #3 (contributed by Lee Morgenstern)
       Find two Pythagorean right triangles A,B,C and D,E,F such that A=D and B=2E. If that is impossible, then find a pair such that A=D and abs(B-2E) is minimum.

Solved by:   Paul Cleary, P.M.A. Hakeem


Pythagorean Triangles #4 (contributed by Lee Morgenstern)
       Find two Pythagorean right triangles A,B,C and D,E,F such that C=D+E and F=A-B. If this is impossible, find the triangles that come closest.

Solved by:   P.M.A. Hakeem, Paul Cleary


## Three Triangles
       Find three Pythagorean right triangles having the same hypotenuse, such that the total area of the first two equals the area of the third, or prove that this is impossible.


## Packing a Cube
       You wish to pack a hollow cube with N spheres so that the least possible space is left over. Either (1) prove that this is achieved by the greedy algorithm, namely at each step you insert the largest possible sphere that will fit, or (2) find the smallest N for which a tighter packing is possible.


Euler Lines
       Let ABC be a general triangle and let O be its circumscribed circle. Let DEF be a second general triangle inscribed in O. Prove that the Euler line of ABC and the Euler line of DEF intersect at O.

Solved by:   Nick McGrath, Fotos Fotiadis


Point on Circumcircle (contributed by Fotos Fotiadis)
       ABC is a triangle whose smallest angle is A. K is a point on the arc BC of the circumcircle. The perpendicular bisectors of AB and AC intersect the line AK at L and M, respectively. The lines BL and CM intersect at T. Prove that BT+CT=AK.


A Point in a Triangle (contributed by Denis Borris )
       A point P is located inside an equilateral triangle with integer side k, such that its distances to the 3 vertices are integers a,b,c. What is the smallest possible value of a? Of k?

Solved by:   Jean Jacquelin, Fotos Fotiadis, P.M.A. Hakeem


Two Points in a Triangle (contributed by Denis Borris )
       Points P and Q are located inside an isosceles triangle with integer sides, such that their distances to the 3 vertices and the length PQ are distinct integers. What is the smallest case?

Solved by:   Jean Jacquelin


* Three Points in a Triangle
       Three points P, Q and R are located inside a triangle ABC with all 15 distances distinct integers. What is the smallest case?

Solved by:   Jean Jacquelin


** Triangle within a Triangle
       A triangle DEF is located inside a triangle ABC with all 15 distances distinct integers. What is the smallest case (smallest area of ABC)?

Solved by:   Jean Jacquelin


Overlap #1
       A circle and a triangle overlap so that the circle cuts each side of the triangle into 3 line segments. What is the smallest possible radius of the circle if the 9 line segments all have distinct integer lengths?

Solved by:   Nick McGrath, Fotos Fotiadis, P.M.A. Hakeem


Overlap #2
       A circle and a triangle overlap so that the circle cuts each side of the triangle into 3 line segments. What is the smallest possible radius of the circle if the 9 line segments and the radius of the circle all have distinct integer lengths?

Solved by:   Nick McGrath


* Overlap #3
       A circle and a triangle overlap so that the center of the circle is at the barycenter of the triangle (where the medians intersect), and the circle cuts each side of the triangle into 3 line segments. What is the smallest possible radius of the circle if the 9 line segments all have distinct integer lengths?

Solved by:   Jean Jacquelin


Pyramid #1
       A pyramid has a quadrilateral base ABCD, vertex V, and altitude VH. What is the smallest such pyramid (in the sense that its longest edge is as small as possible) such that its 8 edges and the line segments AH, BH, CH, DH and VH all have distinct integer lengths?

Solved by:   Jean Jacquelin


Pyramid #2
       A pyramid has a quadrilateral base ABCD, vertex V, and altitude VH, where H is the intersection of AC and BD. What is the smallest such pyramid (in the sense that its longest edge is as small as possible) such that its 8 edges and the line segments AH, BH, CH, DH and VH all have distinct integer lengths?

Solved by:   Jean Jacquelin


Deformed Cube
       Imagine that a cube has been deformed so that it still has 6 planar quadrilateral faces, 8 vertices, and 12 straight edges, but that the 12 edges all have distinct integer lengths. What is the shortest possible length for the longest edge?


* 99 Rods #1 (contributed by Denis Borris )
       You are given 99 thin rigid rods with lengths 1, 2, 3, ..., 99. You are asked to assemble these into as many right triangles as you wish. What is the largest total area that can be obtained? (Each side of a triangle must be one entire rod.)

Solved by:   Lee Morgenstern, P.M.A. Hakeem


* 99 Rods #2 (contributed by Denis Borris )
       You are given 99 thin rigid rods with lengths 1, 2, 3, ..., 99. You are asked to assemble these into as many triangles as you wish, each having an integer area. What is the largest total area that can be obtained? (Each side of a triangle must be one entire rod.)

Solved by:   Lee Morgenstern, P.M.A. Hakeem


* 99 Rods #3
       Repeat problem #1, but a rod may be shared by two right triangles. For example, you could have a 3,4,5 right triangle and a 5,12,13 right triangle sharing the rod of length 5. Your solution must lie flat in the plane without crossings or overlaps. The objective is to cover the largest possible total area. Each side of a triangle must be one entire rod.

Solved by:   Lee Morgenstern


* 99 Rods #4
       Repeat problem #2, but a rod may be shared by two integer triangles. For example, you could have a 13,14,15 integer triangle and a 9,12,15 right triangle sharing the rod of length 15. Your solution must lie flat in the plane without crossings or overlaps. The objective is to cover the largest possible total area. Each side of a triangle must be one entire rod.

Solved by:   Lee Morgenstern


Dissecting a Square
       It is possible to dissect a square into dissimilar right triangles (that is, so no two of the triangles are similar). Find the smallest such square such that all of the triangles have integer sides.

Solved by:   Denis Borris, Fotos Fotiadis


4 Points on a Square
       Let ABCD be a square, and let E, F, G and H be points on the 4 successive sides of the square with the distances AE, EB, BF, FC, CG, GD, DH, HA all integers. What is the smallest such square such that the distances EF, EG, EH, FG, FH, GH are also integers?

Solved by:   Andreas Abraham, Gaurav Agrawal, Denis Borris, Fotos Fotiadis, P.M.A. Hakeem


* 6 Points on a Circle
       What is the smallest possible radius of a circle such that it is possible to place 6 points on the circumference with an integer distance between any two?

Solved by:   Nick McGrath, Denis Borris, Fotos Fotiadis


* 6 Points on a Circle #2
       What is the smallest possible radius of a circle such that it is possible to place 6 points on the circumference with the 15 distances between the points being distinct integers?

Solved by:   Jean Jacquelin


* 6 Points on a Circle #3 (contributed by Denis Borris )
       What is the smallest possible radius of a circle such that it is possible to place 6 points on the circumference with the radius, diameter, and the 15 distances between the points being distinct integers?

Solved by:   Jean Jacquelin


4 Triangles (contributed by Denis Borris )
       Find 4 triangles with integer sides and the same integer area such that 2 of the triangles have 2 equal sides, and the other 2 triangles also have 2 equal sides. That is, A1B1=A2B2,   A1C1=A2C2,   A3B3=A4B4, and A3C3=A4C4. Find the set with the smallest area.

Solved by:   Jean Jacquelin, Fotos Fotiadis, Paul Cleary, P.M.A. Hakeem


* Inscribed Triangle #1
       Let ABC be an arbitrary triangle, and L be an arbitrary line. Is it always possible to find points D on AB, E on BC, and F on AC such that triangle DEF is equilateral, and one of its sides is parallel to L?

Solved by:   Jean Jacquelin


* Inscribed Triangle #2
       Let ABC be an arbitrary triangle, with D a point on AB. Under what circumstances is it possible to find points E on BC and F on AC such that triangle DEF is equilateral?

Solved by:   Jean Jacquelin


* Two Fields (contributed by Denis Borris )
       Farmer Brown has a triangular field with integer sides. Farmer Grey has two square fields with integer sides (measured in meters). Both farmers own the same amount of land (to within one square centimeter), and used the same amount of fencing to enclose their fields. What is the minimum area (in square meters) they could own?

Solved by:   Nick McGrath, Fotos Fotiadis, Paul Cleary


6 Brothers
       The 6 Foodlemyer brothers hate each other with great passion. Today they are on the parade ground, which is a 300-meter square. None of them wants to be anywhere near any of the others. Where should they stand so that the two closest are as far apart as possible?

Solved by:   James Layland, Nick McGrath, Ritwik Chaudhuri, Toby Gottfried, Andreas Abraham, Joshua Woodard, P.M.A. Hakeem


Big Belt
       On Tralfamador they have constructed a new Information Superhighway, a thin flexible belt around the planet's equator in a perfect circle 8,000 miles in diameter. Unfortunately, Foodlemyer Fabricators made the belt 1 inch too long, so they have decided to place a circular disk under the belt at one point to take up the slack.
       How large should the disk be made?

Solved by:   Carlos Rivera, Nick McGrath, Martin Rubin, Mark Rickert, Paul Cleary


Chords (contributed by Sudipta Das)
       Let O be the center of a circle and OR be a radius. Along OR mark off points A1, A2, A3, ..., An. Let the chord perpendicular to OR through Ai meet the circle at Bi.
       Find the smallest circle for which the distances OR, OAi, AiBi, and BiR are all integers for n=1 through n=8.

Solved by:   Nick McGrath, Fotos Fotiadis, P.M.A. Hakeem


Shattern (contributed by Sudipta Das )
       Planet Shattern has installed a forcefield 300 miles wide around the planet to ward off alien invasions. This is a thin flat circular ring in the planet's equatorial plane. Watch-towers have to be installed, at various points on the planet, to guard the forcefield. The authorities have decided to position the watch-towers at those locations which offer the best view of the forcefield (i.e., those positions where the field appears widest). The radius of the planet is 7500 miles and the inner radius of the forcefield is 10200 miles.
       What is the latitude of the watch-towers?

Solved by:   Nick McGrath, Fotos Fotiadis, Gaurav Agrawal, Andreas Abraham


Circumscribe (contributed by Nick McGrath )
       Start with a unit circle. Circumscribe an equilateral triangle around it, then another circle around that. Circumscribe a square around this circle, and another circle around that. Continue with a regular pentagon, hexagon, etc. Does the sequence of circles converge, and if so, what is the limiting value of the radius?

Solved by:   Sudipta Das, Gaurav Agrawal, Fotos Fotiadis, P.M.A. Hakeem


Sum of Circles (contributed by Nick McGrath )
       Start with a unit square. Form an isosceles triangle from two vertices and the midpoint of the opposite side. Inscribe a circle in this triangle. Inscribe a second square in this circle, form a second isosceles triangle as before, and so forth. What is the sum of the areas of all the circles?

Solved by:   Fotos Fotiadis, Mark Rickert, P.M.A. Hakeem


Midpoints of Arcs (contributed by Fotos Fotiadis)
       Two circles intersect at points A and B. Draw a line through B intersecting the two circles at C and D. N is the midpoint of segment CD, P is the midpoint of arc AC and Q is the midpoint of arc AD. Prove that the lines NP and NQ are perpendicular.

Solved by:   Nikolai Dimitrov


In A Square #1 (contributed by Ritwik Chaudhuri)
       There is a point M inside a square ABCD such that angle MAB is 60° and angle MCD is 15°. Find angle MBC.

Solved by:   Rakesh Kumar Banka, Fotos Z. Fotiadis, Sudipta Das, Janaki Mahalingam, Gaurav Agrawal, Arijit Bhattacharyya, Mark Rickert, Paul Cleary, P.M.A. Hakeem


* In A Square #2 (developed from an idea submitted by Sudipta Das)
       There is a square ABCD and a point M in the square such that the distances MA, MB, MC and MD are all integers. Three of these distances are consecutive prime numbers (such as 5, 7, 11). How large is the square?

Solved by:   Ritwik Chaudhuri, Nick McGrath (solved both versions), Sudipta Das, Fotos Fotiadis


What's the Angle? (contributed by Nick McGrath )
       Let ABC be an isosceles triangle with angle BAC=100°. From A draw AD parallel to BC with AD = AB. What is angle ACD?

Solved by:   Sudipta Das, Rakesh Kumar Banka, Janaki Mahalingam, Gaurav Agrawal, Le My An, Arijit Bhattacharyya, Fotos Fotiadis, Diptajit Bhattacharyya, Ritwik Chaudhuri, Mark Rickert, Paul Cleary, P.M.A. Hakeem


* Spider (contributed by Nick McGrath )
       There is a square ABCD with a spider at A and a fly at B. The fly starts walking towards C, while the spider walks directly towards the fly. If the spider walks N times as fast as the fly, and catches the fly at C, what is the value of N? (Extra Credit: what is the equation of the spider's path?)

Solved by:   Sudipta Das, Gaurav Agrawal, Andreas Abraham


5 Circles (contributed by Nick McGrath )
       A, B and C are the integer radii of mutually tangent circles with A>B>C. Circle D is the circumcircle of A, B and C. Circle E is the inscribed circle of A, B and C. Find A, B and C such that: (1) D is the smallest integer possible; (2) D has the smallest value for which both D and E are integers.

Solved by:   Jean Jacquelin, Mark Rickert, Paul Cleary


5 Spheres
       Three solid unit spheres are placed mutually touching. Two identical smaller spheres are placed so they each touch all of first 3 spheres, and touch each other through the central gap. What is the radius of those spheres?

Solved by:   Nick McGrath, Fotos Fotiadis, Andreas Abraham, Gaurav Agrawal, Mark Rickert


TetraSpheres
       Define a tetrasphere to be 4 equal spheres, each externally tangent to the other 3. Nest two tetraspheres so that each sphere in the larger tetrasphere is externally tangent to 3 spheres of the smaller tetrasphere. What is the ratio between the radii of the larger and smaller spheres?

Solved by:   Nick McGrath, Fotos Fotiadis, Gaurav Agrawal, Timothy Cornelius, Joshua Woodard


* Steiner Point
       The Steiner minimal point (also called the Torricelli point) in a triangle is the point at which the sum of the distances to the 3 vertices is minimum. Find a construction for the Steiner minimal point using only a compass.

Solved by:   Jean Jacquelin, Fotos Fotiadis


Trisection (contributed by Ritwik Chaudhuri)
       Let ABC be a triangle, with B a right angle. Let the trisectors of angle C meet AB at D and E, with D closer to A. If AD is 50 and DE is 20, what is the length of EB?

Solved by:   Nick McGrath, Fotos Fotiadis, Sudipta Das, Gaurav Agrawal, Andreas Abraham, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem


Tan (contributed by Nick McGrath )
       Prove that   tan(80) + tan(120) + tan(160) = tan(80) tan(120) tan(160)

Solved by:   Ritwik Chaudhuri, Sudipta Das, Andreas Abraham, Fotos Fotiadis, Gaurav Agrawal, Arijit Bhattacharyya, Diptajit Bhattacharyya, P.M.A. Hakeem


* Block Box #1
       Prove that it is not possible to fill a rectangular box with 2 or more cubes of different sizes. (Note: the sizes are not restricted to integers.)

Solved by:   Nick McGrath, Lee Morgenstern


** Block Box #2
       Is it possible to fill a rectangular box with 2 or more rectangular blocks so that all the dimensions of all the blocks are different? For example, you could not have two blocks whose dimensions are A×B×C and A×D×E.


* Spheres
       There are 4 solid spheres arranged so that each one is touching all of the others. The 3 bottom spheres touch the flat floor at points A, B and C. The top sphere has a radius of 12 centimeters. If it were replaced by a sphere with radius 25 cm, then its center would be 14, 15 and 16 cm further from from points A, B and C, respectively.
       What is the radius of each sphere?

Solved by:   Nick McGrath (solved both the original and corrected versions)


Brick
       Find a rectangular solid where the 12 edges and both diagonals on all 6 faces are integers.
** Extra Credit: Is it possible for the main diagonal to be integral as well?

Solved by:   James Layland, Matthew Ender, Gilles Ravat, Hrishikesh Nene, Rahul Kelkar, Ritwik Chaudhuri, Sudipta Das, Janaki Mahalingam, Gaurav Agrawal, Mark Rickert, Paul Cleary, P.M.A. Hakeem




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