Putnam Archive — Geometry

Suppose that each point in the plane is colored either red or green, subject to the following condition: For every three noncollinear points $A,B,C$ of the same color, the center of the circle passing through $A,B$ and $C$ is also this color. Prove that all points of the plane are the same color.

Let $f\colon [0,1] \to [0, \infty)$ be strictly increasing and continuous. Let $R$ be the region bounded by $x=0$, $x=1$, $y=0$, and $y=f(x)$. Let $x_1$ be the $x$-coordinate of the centroid of $R$. Let $x_2$ be the $x$-coordinate of the centroid of the solid generated by rotating $R$ around the $x$-axis. Prove that $x_1 < x_2$.

Consider a circle $\Omega$ with radius 9 and center at the origin $(0,0)$, and a disc $\Delta$ with radius 1 and center at $(r,0)$, where $0 \leq r \leq 8$. Two points $P$ and $Q$ are chosen independently and uniformly at random on $\Omega$. Which value(s) of $r$ minimize the probability that the chord $\overline{PQ}$ intersects $\Delta$?

Two convex quadrilaterals are called partners if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent? [A diagram has been omitted.]

Let $v_1, \dots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\varepsilon > 0$, there exist integers $a_1,\dots,a_{12}$ such that $\| a_1 v_1 + \cdots + a_{12} v_{12} - v \| < \varepsilon$.

A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021, 2021)$?

Determine all positive integers $N$ for which the sphere $$x^2 + y^2 + z^2 = N$$ has an inscribed regular tetrahedron whose vertices have integer coordinates.

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define $$\rho(x,y) = yh_x - xh_y.$$ Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $\mathcal{S}$ is zero.

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 $\alpha$.

Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.

Suppose that $A,B,C,$ and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient $$\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}$$ is a rational number.

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 $\overrightarrow{PA_2} = \lambda \overrightarrow{PA_1}$.

A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers.

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 $S$.

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 $PB$.

For $m \geq 3$, a list of $\binom{m}{3}$ real numbers $a_{ijk}$ ($1 \leq i < < j < k \leq m$) is said to be area definite for $\mathbb{R}^n$ if the inequality $$\sum_{1 \leq i < j < k \leq m} a_{ijk} \cdot \mathrm{Area}(\Delta A_i A_j A_k) \geq 0$$ holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n$. For example, the list of four numbers $a_{123} = a_{124} = a_{134} = 1$, $a_{234} = -1$ is area definite for $\mathbb{R}^2$. Prove that if a list of $\binom{m}{3}$ numbers is area definite for $\mathbb{R}^2$, then it is area definite for $\mathbb{R}^3$.

Let $d_1, d_2, \dots, d_{12}$ 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.

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P) > 0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P$, then $n > c(P) / V^2$.

Define a growing spiral in the plane to be a sequence of points with integer coordinates $P_0 = (0,0), P_1, \dots, P_n$ such that $n \geq 2$ and:

• the directed line segments $P_0 P_1, P_1 P_2, \dots, P_{n-1} P_n$ are in the successive coordinate directions east (for $P_0 P_1$), north, west, south, east, etc.;
• the lengths of these line segments are positive and strictly increasing.

[Picture omitted.] How many of the points $(x,y)$ with integer coordinates $0\leq x\leq 2011, 0\leq y\leq 2011$ cannot be the last point, $P_n$ of any growing spiral?

Let $G$ be a group, with operation $*$. Suppose that

1. (i)
   $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
2. (ii)
   For each $\mathbf{a},\mathbf{b} \in G$, either $\mathbf{a}\times \mathbf{b} = \mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times \mathbf{b} = 0$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3$.

Prove that $\mathbf{a} \times \mathbf{b} = 0$ for all $\mathbf{a}, \mathbf{b} \in G$.

Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?

Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?

Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \dots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counterclockwise order. Prove that at least one of $f(x)$ and $g(x)$ has degree greater than or equal to $n-1$.

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

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = -1$. (A set $S$ in the plane is called convex if for any two points in $S$ the line segment connecting them is contained in $S$.)

A triangulation $\mathcal{T}$ of a polygon $P$ is a finite collection of triangles whose union is $P$, and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side is a side of exactly one triangle in $\mathcal{T}$. Say that $\mathcal{T}$ is admissible if every internal vertex is shared by 6 or more triangles. For example, [figure omitted.] Prove that there is an integer $M_n$, depending only on $n$, such that any admissible triangulation of a polygon $P$ with $n$ sides has at most $M_n$ triangles.

Find the volume of the region of points $(x,y,z)$ such that $$(x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2).$$

Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.

Show that the curve $x^3 + 3xy + y^3 = 1$ contains only one set of three distinct points, $A$, $B$, and $C$, which are vertices of an equilateral triangle, and find its area.

Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.

Let $n$ be given, $n \geq 4$, and suppose that $P_1, P_2, \dots, P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i$. What is the probability that at least one of the vertex angles of this polygon is acute?

For $i = 1,2$ let $T_i$ be a triangle with side lengths $a_i, b_i, c_i$, and area $A_i$. Suppose that $a_1 \le a_2, b_1 \le b_2, c_1 \le c_2$, and that $T_2$ is an acute triangle. Does it follow that $A_1 \le A_2$?

Determine all real numbers $a > 0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $$R = \{ (x,y) ; 0 \le x \le a, 0 \le y \le f(x) \}$$ has perimeter $k$ units and area $k$ square units for some real number $k$.

Let $n$ be a positive integer, $n \ge 2$, and put $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2 , \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying, in order, $R_1$, then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$.

Let $A,B$, and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a, b, c$ be the distance from $P$ to $A, B, C$, respectively. Show that there is a triangle with side lengths $a, b, c$, and that the area of this triangle depends only on the distance from $P$ to $O$.

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game:

Each player, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex.

Show that the player who signs first will always win by playing as well as possible.

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 $RST$.

Can an arc of a parabola inside a circle of radius 1 have a length greater than 4?

The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon.

Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

Let $B$ be a set of more than $2^{n+1}/n$ distinct points with coordinates of the form $(\pm 1,\pm 1,\ldots,\pm 1)$ in $n$-dimensional space with $n\geq 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

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 $\lim_{\theta\rightarrow 0} |EF|$.

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?

Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.

Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$ whose union contains a set $E\subseteq \mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$ such that $$E\subseteq \cup_{j=1}^n 3D_j.$$ Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb R^2$. Prove that if $$(|AB|+|BC|)^2<8\cdot [ABC]+1$$ then $A, B, C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists.

let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $ABC$ has $H$ as the intersection of the altitudes, $O$ the center of the circumscribed circle, $M$ the midpoint of $BC$, and $F$ the foot of the altitude from $A$. What is the length of $BC$?

The dissection of the 3–4–5 triangle shown below (into four congruent right triangles similar to the original) has diameter $5/2$. Find the least diameter of a dissection of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)

Find the least number $A$ such that for any two squares of combined area 1, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle.

Let $C_1$ and $C_2$ be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find, with proof, the locus of all points $M$ for which there exists points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$.

Let $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$ be the vertices of a convex polygon which contains the origin in its interior. Prove that there exist positive real numbers $x$ and $y$ such that

An ellipse, whose semi-axes have lengths $a$ and $b$, rolls without slipping on the curve $y = c \sin \left( \frac{x}{a} \right)$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$.

Show that if the points of an isosceles right triangle of side length 1 are each colored with one of four colors, then there must be two points of the same color whch are at least a distance $2 - \sqrt{2}$ apart.

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.)

A $2 \times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2, 3)$. It rotates $90^\circ$ clockwise about the point $(2, 0)$. It then rotates $90^\circ$ clockwise about the point $(5, 0)$, then $90^\circ$ clockwise about the point $(7, 0)$, and finally, $90^\circ$ clockwise about the point $(10, 0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is (1,1).

Does there exist an infinite sequence of closed discs $D_1, D_2, D_3, \dots$ in the plane, with centers $c_1, c_2, c_3, \dots$, respectively, such that

1. the $c_i$ have no limit point in the finite plane,
2. the sum of the areas of the $D_i$ is finite, and
3. every line in the plane intersects at least one of the $D_i$?

Does there exist a real number $L$ such that, if $m$ and $n$ are integers greater than $L$, then an $m \times n$ rectangle may be expressed as a union of $4 \times 6$ and $5 \times 7$ rectangles, any two of which intersect at most along their boundaries?

Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to 5/2.

Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?

Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $\overline{L}$ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K, t)$ be the band of points whose distance from $\overline{L}$ is at most $(t/2)w$, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that $$S \cap \bigcap_K B_S(K, t) \neq \emptyset$$ for all $S$? ($K$ runs over all lines in the plane.)

Let $m$ be a positive integer and let $\mathcal{G}$ be a regular $(2m+1)$-gon inscribed in the unit circle. Show that there is a positive constant $A$, independent of $m$, with the following property. For any points $p$ inside $\cal G$ there are two distinct vertices $v_1$ and $v_2$ of $\cal G$ such that $$\left|\,|p-v_1| - |p-v_2|\,\right| < \frac1{m} - \frac{A}{m^3}.$$ Here $|s-t|$ denotes the distance between the points $s$ and $t$.

A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $\displaystyle{\frac{a\sqrt{b} + c}{d}}$, where $a,\,b,\,c,\,d$ are integers.

Label the vertices of a trapezoid $T$ (quadrilateral with two parallel sides) inscribed in the unit circle as $A,\,B,\,C,\,D$ so that $AB$ is parallel to $CD$ and $A,\,B,\,C,\,D$ are in counterclockwise order. Let $s_1,\,s_2$, and $d$ denote the lengths of the line segments $AB,\, CD$, and $OE$, where E is the point of intersection of the diagonals of $T$, and $O$ is the center of the circle. Determine the least upper bound of $\frac{s_1-s_2}{d}$ over all such $T$ for which $d\ne 0$, and describe all cases, if any, in which it is attained.

Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Sketch the region $R$ and find its area.

1. (a)
   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?
2. (b)
   What if “three” is replaced by “nine”?

Curves $A,B,C$ and $D$ are defined in the plane as follows:

Prove that $A \cap B = C \cap D$.

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?

Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T$. Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T$, and the other two sides of $T$, with one side along the side of $R$. For any polygon $X$, let $A(X)$ denote the area of $X$. Find the maximum value, or show that no maximum exists, of $\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and $R,S$ over all rectangles as above.

Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diagonal $pq$. What is the probability that no point of $R$ lies outside of $C$?