Putnam Archive — Easy Putnam Problems

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

Suppose that $P(x) = a_1 x + a_2 x^2 + \cdots + a_n x^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)} = b_0 + b_1 x + b_2 x^2 + \cdots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$.

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 $(2021, 2021)$?

How many positive integers $N$ satisfy all of the following three conditions?

1. (i)
   $N$ is divisible by 2020.
2. (ii)
   $N$ has at most 2020 decimal digits.
3. (iii)
   The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.

Determine all possible values of the expression $$A^3+B^3+C^3-3ABC$$ where $A, B$, and $C$ are nonnegative integers.

Find all ordered pairs $(a,b)$ of positive integers for which $$\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.$$

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 $S$ be the smallest set of positive integers such that

1. (a)
   $2$ is in $S$,
2. (b)
   $n$ is in $S$ whenever $n^2$ is in $S$, and
3. (c)
   $(n+5)^2$ is in $S$ whenever $n$ is in $S$.

Which positive integers are not in $S$?

(The set $S$ is “smallest” in the sense that $S$ is contained in any other such set.)

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

Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer $$p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k}$$ (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.

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.

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 $PB$.

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.

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express $$\sum_{(a,b,c) \in T} \frac{2^a}{3^b 5^c}$$ as a rational number in lowest terms.

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.

A base $10$ over-expansion of a positive integer $N$ is an expression of the form $$N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0$$ with $d_k \neq 0$ 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?

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them.

For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of $$\sum_{n=1}^{2013} c(n) c(n+2).$$

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 $S$ 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 $S$.

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 $h$ and $k$ be positive integers. Prove that for every $\epsilon > 0$, there are positive integers $m$ and $n$ such that $$\epsilon < |h \sqrt{m} - k \sqrt{n}| < 2\epsilon.$$

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 $m$?

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?

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $$\frac{10}{9} = \frac{2!\cdot 5!}{3!\cdot 3! \cdot 3!}.$$ \,

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y) + f(y,z) + f(z,x) = 0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g: \mathbb{R} \to \mathbb{R}$ such that $f(x,y) = g(x) - g(y)$ for all real numbers $x$ and $y$.

Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

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.)

Find all values of $\alpha$ for which the curves $y = \alpha x^2 + \alpha x + \frac{1}{24}$ and $x = \alpha y^2 + \alpha y + \frac{1}{24}$ are tangent to each other.

Let $f$ be a polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n)+1)$ if and only if $n=1$. [Editor's note: one must assume $f$ is nonconstant.]

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

Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that $$\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.$$

Show that every positive integer is a sum of one or more numbers of the form $2^r 3^s$, where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, 23 = 9 + 8 + 6.)

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.)

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 $N$?

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m} \frac{n!}{n^n}.$$

Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, $$n = a_1 + a_2 + \cdots + a_k,$$ with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n=4$ there are four ways: 4, 2+2, 1+1+2, 1+1+1+1.

Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be nonnegative real numbers. Show that

Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k - 1}$ has the form $\frac{P_n(x)}{(x^k - 1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$.

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

Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$.

Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]

Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, $$|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\ 3x+2 & \mbox{if $-1 \leq x \leq 0$} \\ -2x+2 & \mbox{if $x>0$.} \end{cases}$$

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 $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.

A composite (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.