Olympiad problems

1994 Putnam, 2

Tags: Putnam , geometry , conics , ellipse , college contests

Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{x}{2}$, the x-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1$. 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 $\frac{x^2}{9} + y^2 = 1.$

1956 Putnam, B7

Tags: Putnam , algebra , polynomial , roots

The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials $$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$ Prove that $P(z)=Q(z).$

1981 Putnam, B2

Tags: Putnam , inequalities

Determine the minimum value of $$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$ for all real numbers $1\leq r \leq s \leq t \leq 4.$

1991 Putnam, A5

Tags: Putnam , integration , calculus , inequalities , function , college contests

A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have $\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$ Now what?

2014 Postal Coaching, 4

Tags: Putnam , combinatorics unsolved , combinatorics

Show that the number of ordered pairs $(S,T)$ of subsets of $[n]$ satisfying $s>|T|$ for all $s\in S$ and $t>|S|$ for all $t\in T$ is equal to the Fibonacci number $F_{2n+2}$. [color=#008000] Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=296007#p296007 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=515970&hilit=Putnam+1990[/color]

2008 Putnam, A5

Tags: Putnam , algebra , polynomial , rotation , blogs , trigonometry , analytic geometry

Let $ n\ge 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\minus{}1.$

1969 Putnam, B6

Tags: Putnam , matrix multiplication , matrix

Let $A$ and $B$ be matrices of size $3\times 2$ and $2\times 3$ respectively. Suppose that $$AB =\begin{pmatrix} 8 & 2 & -2\\ 2 & 5 &4 \\ -2 &4 &5 \end{pmatrix}.$$ Show that the product $BA$ is equal to $\begin{pmatrix} 9 &0\\ 0 &9 \end{pmatrix}.$

1987 Putnam, A2

Tags: Putnam

The sequence of digits \[ 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots \] is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, $f(1987)$.

2013 India National Olympiad, 4

Tags: Putnam , induction , function , floor function , combinatorics proposed , combinatorics

Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.

2008 Putnam, A4

Tags: Putnam , logarithms , integration , floor function , Putnam calculus

Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?

1946 Putnam, B3

Tags: Putnam , function , geometry , 3D geometry , sphere , physics

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

1950 Putnam, A6

Tags: Putnam

Each coefficient $a_n$ of the power series \[a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots = f(x)\] has either the value of $1$ or the value $0.$ Prove the easier of the two assertions: (i) If $f(0.5)$ is a rational number, $f(x)$ is a rational function. (ii) If $f(0.5)$ is not a rational number, $f(x)$ is not a rational function.

PEN G Problems, 6

Tags: Putnam , irrational number , Irrational numbers

Prove that for any irrational number $\xi$, there are infinitely many rational numbers $\frac{m}{n}$ $\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)$ such that \[\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.\]

1964 Putnam, A4

Tags: Putnam , recurrence relation , periodic

Let $p_n$ be a bounded sequence of integers which satisfies the recursion $$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$ Show that the sequence eventually becomes periodic.

1980 Putnam, B4

Tags: Putnam , set theory

Let $A_1 , A_2 ,\ldots, A_{1066}$ be subsets of a finite set $X$ such that $|A_i | > \frac{1}{2} |X|$ for $1\leq i \leq 1066.$ Prove that there exist ten elements $x_1 ,x_2 ,\ldots , x_{10}$ of $X$ such that every $A_i $ contains at least one of $x_1 , x_2 ,\ldots, x_{10}.$

1969 Putnam, A2

Tags: Putnam , determinant

Let $D_n$ be the determinant of order $n$ of which the element in the $i$-th row and the $j$-th column is $|i-j|.$ Show that $D_n$ is equal to $$(-1)^{n-1}(n-1)2^{n-2}.$$

1969 Putnam, A1

Tags: Putnam , algebra , polynomial

Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$

2023 Putnam, B2

Tags: Putnam , Putnam 2023

For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$?

1987 Putnam, B1

Tags: Putnam

Evaluate \[ \int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}. \]

1973 Putnam, B6

Tags: Putnam , algebra , function , domain , trigonometry , inequalities

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$

2001 Putnam, 4

Tags: Putnam , geometry , vector , geometric transformation , ratio , analytic geometry , symmetry

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ 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 $RST$.

1997 Putnam, 2

Tags: Putnam , floor function , college contests

Players $1,2,\ldots n$ are seated around a table, and each has a single penny. Player $1$ passes a penny to Player $2$, who then passes two pennies to Player $3$, who then passes one penny to player $4$, who then passes two pennies to Player $5$ and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers $n$ for which some player ends up with all the $n$ pennies.

2005 Germany Team Selection Test, 1

Tags: Putnam , inequalities , number theory unsolved , number theory

Find the smallest positive integer $n$ with the following property: For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that \[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]

1964 Putnam, B2

Tags: Putnam , set theory , Elementary

Let $S$ be a set of $n>0$ elements, and let $A_1 , A_2 , \ldots A_k$ be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of $S$ intersects all of the $A_i.$ Prove that $ k=2^{n-1}.$

1956 Putnam, B2

Tags: Putnam , Plane , Subsets

Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).