Putnam Archive — Analysis

Find the largest real number $a$ and the smallest real number $b$ such that $$ax(\pi-x) \leq \sin x \leq bx(\pi-x)$$ for all $x$ in the interval $[0, \pi]$.

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 $x_1 < x_2$.

Let $r_n$ be the $n$th smallest positive solution to $\tan x = x$, where the argument of tangent is in radians. Prove that $$0 < r_{n+1} - r_n - \pi < \frac{1}{(n^2+n)\pi}$$ for $n \geq 1$.

For a real number $a$, let $F_a(x) = \sum_{n \geq 1} n^a e^{2n} x^{n^2}$ for $0 \leq x < 1$. Find a real number $c$ such that

For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.

Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying

1. (a)
   $f(0) > 0$,
2. (b)
   $g(0) = 0$,
3. (c)
   $|f'(x)| \leq |g(x)|$ for all $x$,
4. (d)
   $|g'(x)| \leq |f(x)|$ for all $x$, and
5. (e)
   $f(r) = 0$.

For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:

1. (a)
   $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$;
2. (b)
   $f(t_0) = 1/2$;
3. (c)
   $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$;
4. (d)
   For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$.

Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?

Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \ln(1+x^2)$ in exactly one point.

Find all continuous functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$f(xf(y)) + f(yf(x)) = 1 + f(x+y)$$ for all $x,y > 0$.

For every positive real number $x$, let $$g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}.$$ Find $\lim_{x \to \infty} \frac{g(x)}{x}$.

Let $$I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.$$ Find $$\lim_{R \to \infty} I(R),$$ or show that this limit does not exist.

Determine the maximum value of the sum $$S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}$$ over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying $$\sum_{k=1}^\infty a_k = 1.$$

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.

Let $a_0 = \pi/2$, and let $a_n = \sin(a_{n-1})$ for $n \geq 1$. Determine whether $$\sum_{n=1}^\infty a_n^2$$ converges.

For a positive integer $N$, let $f_N$[Corrected from $F_N$ in the source.] be the function defined by $$f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x).$$ Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$.

For $j \in \{1, 2, 3, 4\}$, let $z_j$ be a complex number with $|z_j| = 1$ and $z_j \neq 1$. Prove that $$3 - z_1 - z_2 - z_3 - z_4 + z_1 z_2 z_3 z_4 \neq 0.$$

Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial $$P(z) = \sum_{k=0}^{2019} b_k z^k.$$ Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy $$1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.$$

Let $f$ be a continuous real-valued function on $\mathbb{R}^3$. Suppose that for every sphere $S$ of radius 1, the integral of $f(x,y,z)$ over the surface of $S$ equals 0. Must $f(x,y,z)$ be identically 0?

Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$. Suppose that for some real number $r>1$, $$\lim_{x \to 0^+} \frac{g(x)}{x^r} = 0.$$ Prove that either $$\lim_{x \to 0^+} g'(x) = 0 \qquad \mbox{or} \qquad \limsup_{x \to 0^+} x^r |g''(x)| = \infty.$$

For all $n \geq 1$, let $$a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.$$ Determine $$\lim_{n \to \infty} \frac{a_n}{n^3}.$$

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):

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 $f$.

Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1,x_2,\dots,x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.

Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$, and $f(x) \geq 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.

Let $n$ be a positive integer, and let $f_n(z) = n + (n-1) z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}\colon |z| \leq 1 \}$.

Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\frac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that $$\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left( \frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0$$ everywhere. Prove that $f$ is one-to-one.

Let $a$ and $b$ be real numbers with $a<b$, and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0, \infty)$ such that $\int_a^b f(x)\,dx = \int_a^b g(x)\,dx$ but $f \neq g$. For every positive integer $n$, define $$I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}\,dx.$$ Show that $I_1, I_2, I_3, \dots$ is an increasing sequence with $\lim_{n \to \infty} I_n = \infty$.

Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.

Evaluate the sum

(As usual, $\ln x$ denotes the natural logarithm of $x$.)

Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that $$\binom{m}{n-1} > \binom{m-1}{n}.$$ Evaluate $$\lim_{n \to \infty} \frac{M(n)}{n}.$$

Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x$$ for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find $$\int_0^1 f(x)\,dx.$$

Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $$\int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|.$$

Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n \geq 0$, $$x_{n+1} = \ln(e^{x_n} - x_n)$$ (as usual, the function $\ln$ is the natural logarithm). Show that the infinite series $$x_0 + x_1 + x_2 + \cdots$$ converges and find its sum.

Find all functions $f$ from the interval $(1, \infty)$ to $(1, \infty)$ with the following property: if $x,y \in (1, \infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$.

Evaluate $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}.$$

For each real number $x$, let $$f(x) = \sum_{n\in S_x} \frac{1}{2^n},$$ where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f + 6f' + 12f'' + 8f'''$ has at least two distinct real zeros.

For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate $$\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.$$

Prove that every nonzero coefficient of the Taylor series of $$(1 - x + x^2)e^x$$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

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

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1 \leq f(x) \leq 1$ for all $x$ and $\int_1^3 f(x)\,dx = 0$. How large can $\int_1^3 \frac{f(x)}{x}\,dx$ be?

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Let $f: [0,1] \to \mathbb{R}$ be a function for which there exists a constant $K>0$ such that $\left| f(x) - f(y) \right| \leq K \left| x - y \right|$ for all $x,y \in [0,1]$. Suppose also that for each rational number $r \in [0,1]$, there exist integers $a$ and $b$ such that $f(r) = a + br$. Prove that there exist finitely many intervals $I_1, \dots, I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1] = \bigcup_{i=1}^n I_i$.

Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form $$f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)$$ for which:

1. (i)
   $f(x) \geq 0$ for all real $x$, and
2. (ii)
   $a_n = 0$ whenever $n$ is a multiple of $3$.

Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and prove that this maximum is attained.

For any continuous real-valued function $f$ defined on the interval $[0,1]$, let

Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1]$, then $$\mathrm{Var}(fg) \leq 2 \mathrm{Var}(f) M(g)^2 + 2 \mathrm{Var}(g) M(f)^2.$$

Let $f: [-1, 1] \to \RR$ be a continuous function such that

• (i)
  $f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
• (ii)
  $f(0) = 1$, and
• (iii)
  $\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.

Prove that $f$ is unique, and express $f(x)$ in closed form.

Let $f(x,y)$ be a continuous, real-valued function on $\RR^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$. Must $f(x,y)$ be identically 0?

Let $S$ be a class of functions from $[0, \infty)$ to $[0, \infty)$ that satisfies:

• (i)
  The functions $f_1(x) = e^x - 1$ and $f_2(x) = \ln(x+1)$ are in $S$;
• (ii)
  If $f(x)$ and $g(x)$ are in $S$, the functions $f(x) + g(x)$ and $f(g(x))$ are in $S$;
• (iii)
  If $f(x)$ and $g(x)$ are in $S$ and $f(x) \geq g(x)$ for all $x \geq 0$, then the function $f(x) - g(x)$ is in $S$.

Prove that if $f(x)$ and $g(x)$ are in $S$, then the function $f(x) g(x)$ is also in $S$.

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 $n > c(P) / V^2$.

Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$ have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)

Prove that, for any two bounded functions $g_1, g_2: \RR \to [1, \infty)$, there exist functions $h_1, h_2: \RR \to \RR$ such that, for every $x \in \RR$, $$\sup_{s \in \RR} (g_1(s)^x g_2(s)) = \max_{t \in \RR} (x h_1(t) + h_2(t)).$$

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for $n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove that $$S = \sum_{n=1}^\infty \frac{1}{a_1...a_n}$$ converges, and evaluate $S$.

Find a real number $c$ and a positive number $L$ for which $$\lim_{r\to\infty} \frac{r^c \int_0^{\pi/2} x^r \sin x \,dx}{\int_0^{\pi/2} x^r \cos x \,dx} = L.$$

Let $F : \RR^2 \to \RR$ and $g : \RR \to \RR$ be twice continuously differentiable functions with the following properties:

• $F(u,u) = 0$ for every $u \in \RR$;
• for every $x \in \RR$, $g(x) > 0$ and $x^2 g(x) \leq 1$;
• for every $(u,v) \in \RR^2$, the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u), -g(v) \rangle$.

Prove that there exists a constant $C$ such that for every $n\geq 2$ and any $x_1,\dots,x_{n+1} \in \RR$, we have $$\min_{i \neq j} |F(x_i,x_j)| \leq \frac{C}{n}.$$

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at 0?

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

Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that $$f'(x) = \frac{f(x+n)-f(x)}{n}$$ for all real numbers $x$ and all positive integers $n$.

Suppose that the function $h:\mathbb{R}^2\to \mathbb{R}$ has continuous partial derivatives and satisfies the equation $$h(x,y) = a \frac{\partial h}{\partial x}(x,y) + b \frac{\partial h}{\partial y}(x,y)$$ for some constants $a,b$. Prove that if there is a constant $M$ such that $|h(x,y)|\leq M$ for all $(x,y) \in \mathbb{R}^2$, then $h$ is identically zero.

Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty} f(x) = 0$. Prove that $\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that $$a_1^m + a_2^m + a_3^m + \cdots = m$$ for every positive integer $m$?

Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?

Functions $f,g,h$ are differentiable on some open interval around $0$ and satisfy the equations and initial conditions

Find an explicit formula for $f(x)$, valid in some open interval around $0$.

Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example, $$d_3 = \left| \begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|.$$ The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$.

Let $f:[0,1]^2 \to \mathbb{R}$ be a continuous function on the closed unit square such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are continuous on the interior $(0,1)^2$. Let $a = \int_0^1 f(0,y)\,dy$, $b = \int_0^1 f(1,y)\,dy$, $c = \int_0^1 f(x,0)\,dx$, $d = \int_0^1 f(x,1)\,dx$. Prove or disprove: There must be a point $(x_0,y_0)$ in $(0,1)^2$ such that $$\frac{\partial f}{\partial x} (x_0,y_0) = b - a \quad \mbox{and} \quad \frac{\partial f}{\partial y} (x_0,y_0) = d - c.$$