Order & Extremal Elements
If a collection of things can be ordered, consider ordering it.
If that ordered collection has
a minimum or maximum member,
it may be a good idea to examine that member.
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An odd number of people are standing in a room,
each one of them wielding a pie,
and each one looking towards their nearest neighbor.
All at once, each person tosses their pie
into the face of their nearest neighbour.
Prove that somebody ends up without pie on their face.
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For some positive integer \(n\),
there are \(2n\) distinct points in the plane, no three collinear.
Half of them are colored blue and the other half are colored orange.
Show that it is possible to pair each blue dot with an orange dot in such a way
that no two of the \(n\) line segments connecting those pairs intersect.
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Putnam and Beyond
Prove that for any polygon in the plane,
there exists a vertex of that polygon
and an edge of that polygon not containing that vertex,
such that the projection of the vertex onto
the line containing (extending) that edge
lies on the edge itself.
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Putnam and Beyond
The system of roadways in the country of Talpapiopia are very peculiar.
All roads between two cities are one-way,
and are set up in such a way that once you leave a city
you can never return to it via the roadways again.
Prove that there exists a city into which all roads enter
and a city from which all roads exit.
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Let \(S\) be a finite set of three or more points in the plane
with the property that any three points in \(S\)
can be covered by a triangle of area one.
Show that the entire set \(S\)
can be covered by a triangle of area four.
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Find the smallest number of chips
which can be placed in the squares of an \(n \times n\) grid
in such a way that for each coordinate \((i,j)\) for which
there is no chip in the \((i,j)\mathrm{th}\) square,
the sum total number of chips in the \(i\mathrm{th}\) row
together with the chips in the \(j\mathrm{th}\) column
is at least equal to \(n\).
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Putnam and Beyond
Consider a finite set of spherical planets in space, all of the same size.
On the surface of each planet consider the region of all points
not visible from any other planet.
Prove that the sum of the areas of these regions over all the planets
is equal to the surface area of a single planet.
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Putnam and Beyond
Given finitely many squares whose areas add up to one,
show that they can be arranged without overlaps
inside a square of area two.
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Putnam and Beyond
Given \(n \gt 3\) points in the plane,
prove that three points form an angle
less than or equal to \(\pi/n.\)
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Let \(d_1, d_2, \dotsc, d_{12}\) be real numbers in the open interval \((1,12)\).
Show that there exist distinct indices \(i, j, k\)
such that \(d_i, d_j, d_k\) are the side lengths of an acute triangle.