Olympiad problems

1981 Putnam, B4

Let $V$ be a set of $5\times7$ matrices, with real entries and closed under addition and scalar multiplication. Prove or disprove the following assertion: If $V$ contains matrices of ranks $0, 1, 2, 4,$ and $5$, then it also contains a matrix of rank $3$.

1969 Putnam, B4

Tags: Putnam , curves

Show that any curve of unit length can be covered by a closed rectangle of area $1 \slash 4$.

1996 Putnam, 2

Tags: Putnam , college contests

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ 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 $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.

1993 Putnam, A3

Tags: Putnam , combinatorics

Let $P$ be the set of all subsets of ${1, 2, ... , n}$. Show that there are $1^n + 2^n + ... + m^n$ functions $f : P \longmapsto {1, 2, ... , m}$ such that $f(A \cap B) = min( f(A), f(B))$ for all $A, B.$

2022 Putnam, A5

Tags: Putnam , Putnam 2022

Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

1994 Putnam, 5

Tags: Putnam , limit , college contests

Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

1948 Putnam, B6

Tags: Putnam , geometry , 3D geometry , complex numbers , linear algebra , matrix

Answer wither (i) or (ii): (i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$ (ii) Let $(a_{ij})$ be a matrix such that $$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$ for all $i.$ Show that the determinant is not equal to $0.$

2011 Putnam, B5

Tags: Putnam , integration , function , inequalities , calculus , Support , vector

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

1954 Putnam, A1

Tags: Putnam , linear algebra , matrix , permutation

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

1994 Putnam, 6

Tags: Putnam , function , college contests

Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions \[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ \cdots \circ f_{10}^{\epsilon_{10}}\} \] where $\epsilon _i=0$ or $1$ for $1\le i\le 10.\; (f_i^{0}$ is the identity function and $f_i^1=f_i)$. Show that if $A$ is a finite set of integers then at most $512$ of the functions in $\mathcal{F}$ map $A$ into itself.

2018 Putnam, A6

Tags: Putnam , Putnam 2018

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.

1992 Putnam, A4

Tags: function , calculus , derivative , limit , trigonometry , search , Putnam

Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$

2014 Putnam, 3

Tags: Putnam , limit , induction , inequalities , algebra , difference of squares , Putnam 2014

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

1988 Putnam, A3

Tags: Putnam

Determine, with proof, the set of real numbers $x$ for which \[ \sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x \] converges.

1997 Putnam, 3

Tags: Putnam , number theory , greatest common divisor , college contests

For each positive integer $n$ write the sum $\sum_{i=}^{n}\frac{1}{i}=\frac{p_n}{q_n}$ with $\text{gcd}(p_n,q_n)=1$. Find all such $n$ such that $5\nmid q_n$.

1954 Putnam, B1

Tags: Putnam , number theory

Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.

1953 Putnam, A6

Tags: Putnam , Sequences , limit

Show that the sequence $$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$ converges and evaluate the limit.

PEN D Problems, 2

Tags: modular arithmetic , Putnam , group theory , abstract algebra , blogs , Congruences

Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]

1997 Putnam, 4

Tags: Putnam , function , college contests

Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

1948 Putnam, B5

Tags: Putnam , geometry , area

The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?

1978 Putnam, B3

Tags: Putnam , Polynomials , Recurrence , Sequences

The sequence $(Q_{n}(x))$ of polynomials is defined by $$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$ and for $m \geq 1 $ by $$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$ $$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$ Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$

1965 Putnam, A3

Tags: Putnam , limit

Show that, for any sequence $a_1,a_2,\ldots$ of real numbers, the two conditions \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha \] and \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha \] are equivalent.

1965 Putnam, A1

Tags: Putnam , college contests

Let $ ABC$ be a triangle with angle $ A <$ angle $ C < 90^\circ <$ angle $ B$. Consider the bisectors of the external angles at $ A$ and $ B$, each measured from the vertex to the opposoite side (extended). Suppose both of these line-segments are equal to $ AB$. Compute the angle $ A$.

2021 Putnam, A6

Tags: Putnam , Putnam 2021

Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as the product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?

1961 Putnam, A1

Tags: Putnam , analytic geometry , Intersection of set

The graph of the equation $x^y =y^x$ in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.