Synthetic Geometry

  1. MIT

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

  3. MIT

    For an equilateral triangle and a point xx on its interior, let aa and bb and cc be the perpendicular distances from xx to each of the three sides of the triangle. Classify all points xx on the interior of the triangle, such that the sum a+b+ca+b+c is minimized,

  4. Nick’s Mathematical Puzzles ☆☆☆

    Two similar triangles with integral side-lengths have two pairs of their sides the same. If the third sides differ by 20141,20141, find the lengths of all the sides.

  5. Putnam & Beyond

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

  6. 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?

  7. James Stewart

    Each edge of a cubical box has length one. The box contains nine spherical balls with the same radius r.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.r.

  8. James Stewart

    Consider a solid box with length L,L, width W,W, and height H.H. Let SS 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 SS in terms of L,L, W,W, and H.H.

  9. Putnam 2017 B1

    Let L1L_1 and L2L_2 be distinct lines in the plane. Prove that L1L_1 and L2L_2 intersect if and only if, for every real number λ≠0\lambda\neq 0 and every point PP not on L1L_1 or L2,L_2, there exist points A1A_1 on L1L_1 and A2A_2 on L2L_2 such that PA2→=λPA1→.\overrightarrow{PA_2} = \lambda \overrightarrow{PA_1}.

  10. Putnam 2015 A1

    Let AA and BB be points on the same branch of the hyperbola xy=1xy=1. Suppose that PP is a point lying between AA and BB on this hyperbola, such that the area of the triangle APBAPB is as large as possible. Show that the region bounded by the hyperbola and the chord APAP has the same area as the region bounded by the hyperbola and the chord PBPB.

  11. Putnam 1986 B1

    Inscribe a rectangle of base bb and height hh 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 hh do the rectangle and triangle have the same area?

  12. 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?

  13. Putnam 1999 B1

    Right triangle ABCABC has right angle at CC and ∠BAC=θ;\angle BAC =\theta; the point DD is chosen on ABAB so that ∣AC∣=∣AD∣=1;|AC|=|AD|=1; the point EE is chosen on BCBC so that ∠CDE=θ.\angle CDE = \theta. The perpendicular to BCBC at EE meets ABAB at FF. Evaluate lim⁡θ→0∣EF∣.\lim_{\theta\rightarrow 0} |EF|.

  14. Putnam 2008 B1

    What is the maximum number of rational points that can lie on a circle in R2\mathbf{R}^2 whose center is not a rational point? (A rational point is a point both of whose coordinates are rational numbers.)

  15. Paul Erdős ($50)

    Let AA and BB be disjoint subsets of R2\mathbf{R}^2 of size nn and n−3n-3 respectively, such that the points of AA are not contained within a single line. Must there necessarily always be a line in R2\mathbf{R}^2 containing at least two points from AA and no points from B?B?

  16. Putnam 2016 B3

    Suppose that SS is a finite set of points in the plane such that the area of triangle △ABC\triangle ABC is at most 11 whenever A,A, B,B, and CC are in SS. Show that there exists a triangle of area 44 that (together with its interior) covers the set SS.

  17. 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?

  18. Putnam 2019 A2

    In the triangle △ABC\triangle ABC, let GG be the centroid, and let II be the center of the inscribed circle. Let α\alpha and β\beta be the angles at the vertices AA and BB, respectively. Suppose that the segment IGIG is parallel to ABAB and that β=2tan⁡−1(1/3)\beta = 2 \tan^{-1} (1/3). Find α\alpha.

  19. Putnam 2001 A4

    Triangle ABCABC has an area 1. Points E,F,GE,F,G lie, respectively, on sides BCBC, CACA, ABAB such that AEAE bisects BFBF at point R,R, BFBF bisects CGCG at point S,S, and CGCG bisects AEAE at point T.T. Find the area of the triangle RST.RST.