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D29. Geometric-Trigonometric Inequality
Posted: 2025 Dec 23 Solutions will be accepted through 2025 Jan 2Since no respondents provided a solution to his last problem, Prof. Aliyev is willing to offer one more chance to unravel his mysterious diagrams. In the accompanying figure, nonintersecting circles with radii r < R have commonexternal tangent lines A B and C D , where the named points arethe points of tangency. Choose point…
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D28. Tile Troubles
Posted: 2025 May 12 Solutions will be accepted through 2025 Jul 15Dr. Arsalan Wares’ letter this month is beautiful, as always, but sadly recounts a tale of honeycomb horror. The diagram in Figure 3 depicts a portion of a tiling of the plane by regular hexagonal tiles. All line segments shown have endpoints at the vertices of the tiling. Dr. Wares woke up one morning to…
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D27. Penned Out
Posted: 2025 May 12 Solutions will be accepted through 2025 Jul 15Figure 1 (below left) shows a perfect two-point perspective diagram of Farmer Nell’s red-roofed barn and precisely square pigpen in front of it. (“Two-point perspective” means that all lines that are vertical in three dimensions appear perfectly vertical in the diagram, while any two parallel horizontal lines intersect on the horizon line N E .)…
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D26. Aperiodic Partition
Posted: 2025 May 12 Solutions will be accepted through 2025 Jul 15(with thanks to Kate Stange, University of Colorado) Recall that an (infinite) arithmetic progression is the set of all numbers of the form a + n d , where a and d are fixed and n is any natural number. For example, 5, 8, 11, 14, … , and so on forever, is an arithmetic…
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D25. Whodunni’s Shuffle
Posted: 2025 May 12 Solutions will be accepted through 2025 Jul 15Our frequent contributor Prof. Stewart is back with a bit of permutation prestidigitation: The great stage magician Harry Whodunni shuffles a pack of (more than two) cards in the following simple way: he moves the top card to the next-to-bottom position, and then turns the whole pack over (which has the side effect of turning…
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D24. Sliding Ratios
Posted: 2024 Sep 30 Solutions will be accepted through 2025 Jul 15As shown in Figure 4, A B C is a triangle. Points F and E lie on side A B , with F closer to B . Point D lies on side A C . Point G is the intersection of segments B D and C E , and H is the intersection of ray…
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D23. Factor Network
Posted: 2024 Sep 30You want to label the vertices (corners) of a regular octahedron (see Figure 3) with whole numbers bigger than one so that the labels of two vertices have a common factor (bigger than one) if and only if they are adjacent on the octahedron (connected by an edge). Is it possible? If so, what is…
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D22. Fill the Jar
Posted: 2024 Sep 10You are making sugar cookies from a batch of dough in the following way: You take your dough and roll it into a perfect square of just the right thickness (you are a very precise baker). Then you cut out 1-inch radius, perfectly circular, cookies arranged in a square grid so that the cookies are…
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D21. Light Chase
Posted: 2024 Sep 10You have a puzzle toy which consists of a ring of lit buttons, each of which can glow red, yellow, or (bluish-)green, as illustrated in Figure 1. When you push a button, its color cycles in the order red → green → yellow → red. The catch is, the lights two positions away in each…
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D20. Geometry Game
Posted: 2024 Mar 01You and a buddy start with a rectangular cake, represented by rectangle A B C D in Figure 1. Then you get to pick any point E between C and D , and the cake is sliced with two straight cuts, from B to E and E to A . Then it’s your friend’s turn…