Found problems: 966
2023 Putnam, A2
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2 n$; that is to say, $p(x)=$ $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$ for some real coefficients $a_0, \ldots, a_{2 n-1}$. Suppose that $p(1 / k)=k^2$ for all integers $k$ such that $1 \leq|k| \leq n$. Find all other real numbers $x$ for which $p(1 / x)=x^2$.
2011 Putnam, B6
Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$
\[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]
2001 Putnam, 5
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
2015 Putnam, B2
Given a list of the positive integers $1,2,3,4,\dots,$ take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16.$ Continue in this way, crossing off the three smallest remaining numbers and their sum and consider the sequence of sums produced: $6,16,27, 36, \dots.$ Prove or disprove that there is some number in this sequence whose base 10 representation ends with $2015.$
2007 Putnam, 6
A [i]triangulation[/i] $ \mathcal{T}$ of a polygon $ P$ is a finite collection of triangles whose union is $ P,$ and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of $ P$ is a side of exactly one triangle in $ \mathcal{T}.$ Say that $ \mathcal{T}$ is [i]admissible[/i] if every internal vertex is shared by $ 6$ or more triangles. For example
[asy]
size(100);
dot(dir(-100)^^dir(230)^^dir(160)^^dir(100)^^dir(50)^^dir(5)^^dir(-55));
draw(dir(-100)--dir(230)--dir(160)--dir(100)--dir(50)--dir(5)--dir(-55)--cycle);
pair A = (0,-0.25);
dot(A);
draw(A--dir(-100)^^A--dir(230)^^A--dir(160)^^A--dir(100)^^A--dir(5)^^A--dir(-55)^^dir(5)--dir(100));
[/asy]
Prove that there is an integer $ M_n,$ depending only on $ n,$ such that any admissible triangulation of a polygon $ P$ with $ n$ sides has at most $ M_n$ triangles.
1954 Putnam, B5
Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$
Prove that
$$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$
1960 Putnam, B4
Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.
2023 Putnam, B6
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2, \ldots, n\}$, let $s(i, j)$ be the number of pairs $(a, b)$ of nonnegative integers satisfying $a i+b j=n$. Let $S$ be the $n$-by-n matrix whose $(i, j)$-entry is $s(i, j)$.
For example, when $n=5$, we have $S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]$.
Compute the determinant of $S$.
Putnam 1939, B5
Do either $(1)$ or $(2)$:
$(1)$ Prove that $\int_{1}^{k} [x] f'(x) dx = [k] f(k) - \sum_{1}{[k]} f(n),$ where $k > 1,$ and $[z]$ denotes the greatest integer $\leq z.$ Find a similar expression for: $\int_{1}^{k} [x^2] f'(x) dx.$
$(2)$ A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from $1,000 \dfrac{ft}{s}$ to $900 \dfrac{ft}{s}$ over $1200 ft.$ Find the time taken to the nearest $0.01 s.$ [No calculators or log tables allowed!]
1942 Putnam, A1
A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices
are always on the $x$- and $y$-axes respectively. Find the locus of the midpoint of the square.
2015 Putnam, A1
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 $PB.$
1961 Putnam, B5
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define
$$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$
Prove either part of the inequality
$$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$
where the middle term has $k$ factorial symbols.
1975 Putnam, A3
Let $0<\alpha<\beta <\gamma\in \mathbb{R}$. Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}$. Define $f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$. At what points of $K$ does $f$ assume its minimal and maximal values?
1993 Putnam, A6
Let $a_0, a_1, a_2, ...$ be a sequence such that: $a_0 = 2$; each $a_n = 2$ or $3; a_n =$the number of $3$s between the $n$th and $n+1$th $2$ in the sequence. So the sequence starts: $233233323332332 ...$ . Show that we can find $\alpha$ such that $a_n = 2$ iff $n = [\alpha m]$ for some integer $m \geq 0$.
1968 Putnam, A4
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
2004 Putnam, A4
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2\cdots x_n$ can be expressed identically in the form
\[x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n\]
where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers, $-1,0,1.$
1975 Putnam, B4
Does a circle have a subset which is topologically closed and which contains exactly one point of each pair of diametrically opposite points?
1961 Putnam, B7
Given a sequence $(a_n)$ of non-negative real numbers such that $a_{n+m}\leq a_{n} a_{m} $ for all pairs of positive integers $m$ and $n,$ prove that the sequence $(\sqrt[n]{a_n })$ converges.
1957 Putnam, B2
In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$
2000 Putnam, 6
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
1959 Putnam, B4
Given the following matrix
$$\begin{pmatrix}
11& 17 & 25& 19& 16\\
24 &10 &13 & 15&3\\
12 &5 &14& 2&18\\
23 &4 &1 &8 &22 \\
6&20&7 &21&9
\end{pmatrix},$$
choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.
2016 Putnam, A3
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that
\[f(x)+f\left(1-\frac1x\right)=\arctan x\]
for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find
\[\int_0^1f(x)\,dx.\]
1985 Putnam, A1
Determine, with proof, the number of ordered triples $\left(A_{1}, A_{2}, A_{3}\right)$ of sets which have the property that
(i) $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$ and
(ii) $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$
Express your answer in the form $2^{a} 3^{b} 5^{c} 7^{d},$ where $a, b, c, d$ are nonnegative integers.
1986 Putnam, B2
Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$.
1955 Putnam, B4
Do there exist $1,000,000$ consecutive integers each of which contains a repeated prime factor?