Synthetic Geometry

  1. MIT

    Given two line segments of lengths \(x\) and \(y,\) describe a simple geometric construction for constructing a segment of length \(\sqrt{xy}.\)
  2. What is the volume of an icosahedron of side-length one?

  3. MIT

    For an equilateral triangle and a point \(x\) on its interior, let \(a\) and \(b\) and \(c\) be the perpendicular distances from \(x\) to each of the three sides of the triangle. Classify all points \(x\) on the interior of the triangle, such that the sum \(a+b+c\) is minimized,

  4. Putnam & Beyond

    Prove the plane cannot be covered by the interiors of finitely many parabolas.

  5. MIT

    Lines are drawn from the vertices of a square to the midpoints of the sides as shown. What is the ratio of the area of the original square to the area of the center square? Can you solve the problem without making any arithmetic or algebraic calculations?

  6. James Stewart

    Each edge of a cubical box has length one. The box contains nine spherical balls with the same radius \(r.\) The center of one ball is at the center of the cube and it touches all of the other eight balls. Each of the other eight balls touches three sides of the box. Thus, the balls are tightly packed in the box. Find \(r.\)

  7. James Stewart

    Consider a solid box with length \(L,\) width \(W,\) and height \(H.\) Let \(S\) be the set of all points that are a distance of at most one from some point on the surface of the box. Express the volume of \(S\) in terms of \(L,\) \(W,\) and \(H.\)

  8. Putnam 2017 B1

    Let \(L_1\) and \(L_2\) be distinct lines in the plane. Prove that \(L_1\) and \(L_2\) intersect if and only if, for every real number \(\lambda\neq 0\) and every point \(P\) not on \(L_1\) or \(L_2,\) there exist points \(A_1\) on \(L_1\) and \(A_2\) on \(L_2\) such that \(\overrightarrow{PA_2} = \lambda \overrightarrow{PA_1}.\)

  9. Putnam 2015 A1

    Let \(A\) and \(B\) be points on the same branch of the hyperbola \(xy=1\). Suppose that \(P\) is a point lying between \(A\) and \(B\) on this hyperbola, such that the area of the triangle \(APB\) is as large as possible. Show that the region bounded by the hyperbola and the chord \(AP\) has the same area as the region bounded by the hyperbola and the chord \(PB\).

  10. Putnam 1986 B1

    Inscribe a rectangle of base \(b\) and height \(h\) in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of \(h\) do the rectangle and triangle have the same area?

  11. Putnam 1998 A1

    A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?

  12. Putnam 1999 B1

    Right triangle \(ABC\) has right angle at \(C\) and \(\angle BAC =\theta;\) the point \(D\) is chosen on \(AB\) so that \(|AC|=|AD|=1;\) the point \(E\) is chosen on \(BC\) so that \(\angle CDE = \theta.\) The perpendicular to \(BC\) at \(E\) meets \(AB\) at \(F\). Evaluate \(\lim_{\theta\rightarrow 0} |EF|.\)

  13. Putnam 2008 B1

    What is the maximum number of rational points that can lie on a circle in \(\mathbf{R}^2\) whose center is not a rational point? (A rational point is a point both of whose coordinates are rational numbers.)

  14. Paul Erdős ($50)

    Let \(A\) and \(B\) be disjoint subsets of \(\mathbf{R}^2\) of size \(n\) and \(n-3\) respectively, such that the points of \(A\) are not contained within a single line. Must there necessarily always be a line in \(\mathbf{R}^2\) containing at least two points from \(A\) and no points from \(B?\)

  15. Putnam 2016 B3

    Suppose that \(S\) is a finite set of points in the plane such that the area of triangle \(\triangle ABC\) is at most \(1\) whenever \(A,\) \(B,\) and \(C\) are in \(S\). Show that there exists a triangle of area \(4\) that (together with its interior) covers the set \(S\).

  16. Putnam 1988 A4

    If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? What if three is replaced by nine?

  17. Putnam 2019 A2

    In the triangle \(\triangle ABC\), let \(G\) be the centroid, and let \(I\) be the center of the inscribed circle. Let \(\alpha\) and \(\beta\) be the angles at the vertices \(A\) and \(B\), respectively. Suppose that the segment \(IG\) is parallel to \(AB\) and that \(\beta = 2 \tan^{-1} (1/3)\). Find \(\alpha\).

  18. Putnam 2001 A4

    Triangle \(ABC\) has an area 1. Points \(E,F,G\) lie, respectively, on sides \(BC\), \(CA\), \(AB\) such that \(AE\) bisects \(BF\) at point \(R,\) \(BF\) bisects \(CG\) at point \(S,\) and \(CG\) bisects \(AE\) at point \(T.\) Find the area of the triangle \(RST.\)