Olympiad problems

1978 Putnam, B5

Find the largest $a$ for which there exists a polynomial $$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$ with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$

2004 Putnam, B3

Tags: Putnam , function , geometry , perimeter , college contests

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 $k$.

2004 Putnam, A1

Tags: Putnam , induction , inequalities , percent , college contests , Putnam easy

Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N),$ of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than 80% of $N,$ but by the end of the season, $S(N)$ was more than 80% of $N.$ Was there necessarily a moment in between when $S(N)$ was exactly 80% of $N$?

1954 Putnam, B2

Tags: Putnam , series , harmonic series , limit

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that $$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$ \text{ii}) \; S\ne s.$$

2014 Putnam, 2

Tags: Putnam , linear algebra , matrix , induction , college contests , Putnam 2014 , Putnam matrices

Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

1981 Putnam, A5

Tags: Putnam , polynomial , roots

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

1941 Putnam, B2

Tags: Putnam , Convergence , series

Find (i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$. (ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$. (iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.

1954 Putnam, A4

Tags: Putnam , angle , physics

A uniform rod of length $2k$ and weight $w$ rests with the end $A$ against a vertical wall, while the lower end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A.$ If the system is in equilibrium, determine the angle $ABC.$

1946 Putnam, B4

Tags: Putnam , Sequences , approximation

For each positive integer $n$, put $$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$

2014 Putnam, 1

Tags: Putnam , modular arithmetic , induction , college contests , Putnam 2014

A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?

1992 Putnam, B2

Tags: factorial , Putnam , Stanford , college

For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that $Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$. [hide="hint"] Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]

2009 Putnam, B5

Tags: Putnam , function , limit , integration , algebra , domain , geometry

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

1952 Putnam, A3

Tags: Putnam

Develop necessary and sufficient conditions which ensure that $r_1, r_2, r_3$ and $r_1^2, r_2^2, r_3^2$ are simultaneously roots of the equation $x^3 + ax^2 + bx + c = 0.$

1940 Putnam, B4

Tags: Putnam , geometry , 3D geometry , sphere

Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface $$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$ is the sphere $$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

1963 Putnam, B2

Tags: Putnam , dense

Let $S$ be the set of all numbers of the form $2^m 3^n$, where $m$ and $n$ are integers. Is $S$ dense in the set of positive real numbers?

1952 Putnam, A2

Tags: Putnam

Show that the equation \[ (9 - x^2) \left (\frac{\mathrm dy}{\mathrm dx} \right)^2 = (9 - y^2)\] characterizes a family of conics touching the four sides of a fixed square.

1998 Putnam, 1

Tags: Putnam , geometry , 3D geometry , rectangle , college contests , Putnam easy

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?

1960 Putnam, B2

Tags: Putnam , infinite series , series

Evaluate the double series $$\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3k -j -(k+j)^{2}}.$$

1965 Putnam, B1

Tags: Putnam , limit , integration , trigonometry , calculus , college contests

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

1942 Putnam, A4

Tags: Putnam , conics

Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin?

2009 Putnam, A2

Tags: Putnam , function , calculus , derivative , trigonometry , logarithms , integration

Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

2019 Putnam, B4

Tags: Putnam , Putnam 2019

Let $\mathcal F$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq 1$, $y\geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives): \[xf_x + yf_y = xy\ln(xy),\] \[x^2f_{xx} + y^2f_{yy} = xy.\] For each $f\in\mathcal F$, let \[ m(f) = \min_{s\geq 1}\left(f(s+1,s+1) - f(s+1,s)-f(s,s+1) + f(s,s)\right). \] Determine $m(f)$, and show that it is independent of the choice of $f$.

1978 Putnam, B6

Tags: Putnam , Summation , inequalities

Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that $$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$ where each summation is over all admissible ordered pairs $(h,k).$

2014 Contests, 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.

Putnam 1939, A1

Tags: Putnam

Let $C$ be the curve $y^2 = x^3$ (where $x$ takes all non-negative real values). Let $O$ be the origin, and $A$ be the point where the gradient is $1.$ Find the length of the curve from $O$ to $A.$