Found problems: 966
1959 Putnam, A6
Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that
(i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and
(ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$
2017 Putnam, B6
Find the number of ordered $64$-tuples $\{x_0,x_1,\dots,x_{63}\}$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[x_0+x_1+2x_2+3x_3+\cdots+63x_{63}\]
is divisible by $2017.$
1992 Putnam, B5
Let $D_n$ denote the value of the $(n -1) \times (n - 1)$ determinant
$$ \begin{pmatrix}
3 & 1 &1 & \ldots & 1\\
1 & 4 &1 & \ldots & 1\\
1 & 1 & 5 & \ldots & 1\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & 1 & 1 & \ldots & n+1
\end{pmatrix}.$$
Is the set $\left\{ \frac{D_n }{n!} \, | \, n \geq 2\right\}$ bounded?
2022 Putnam, B4
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1, \ldots, x_n$ such that each of the sets $$\{x_1, x_2, x_3\}, \{x_2, x_3, x_4\},\ldots,\{x_{n-2}, x_{n-1}, x_n\}, \{x_{n-1}, x_n, x_1\},\text{ and } \{x_n, x_1, x_2\}$$ forms a 3-term arithmetic progression when arranged in increasing order.
1986 Putnam, B3
Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.
1986 Putnam, A6
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
1958 February Putnam, A5
Show that the integral equation
$$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$
has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$
1947 Putnam, B5
Let $a,b,c,d$ be distinct integers such that
$$(x-a)(x-b)(x-c)(x-d) -4=0$$
has an integer root $r.$ Show that $4r=a+b+c+d.$
2007 Moldova Team Selection Test, 4
We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.
2015 Putnam, A4
For each real number $x,$ let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] 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 $f(x)\ge L$ for all $x\in [0,1)$?
(As usual, $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$
Russian TST 2019, P1
Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient
\[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\]
is a rational number.
1998 Putnam, 2
Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.
2000 Putnam, 4
Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.
1946 Putnam, A1
Suppose that the function $f(x)=a x^2 +bx+c$, where $a,b,c$ are real, satisfies the condition $|f(x)|\leq 1$ for $|x|\leq1$. Prove that $|f'(x)|\leq 4$ for $|x|\leq1$.
2019 Putnam, B3
Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.
Putnam 1938, A4
A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through the axis of the cylinder. The angle between the two half-planes is $\theta$. Prove that the volume of the notch is minimized (for given tree and $\theta$) by taking the bounding planes at equal angles to the horizontal plane.
1961 Putnam, A2
For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.
1948 Putnam, B2
A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.
2010 Contests, A2
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 $n.$
1995 Putnam, 2
For what pairs of positive real numbers $(a,b)$ does the improper integral $(1)$ converge?
\begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}
1947 Putnam, A5
Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define
$$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$
Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.
1946 Putnam, A6
A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$, where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.
1970 Putnam, A2
Consider the locus given by the real polynomial equation
$$ Ax^2 +Bxy+Cy^2 +Dx^3 +E x^2 y +F xy^2 +G y^3=0,$$
where $B^2 -4AC <0.$ Prove that there is a positive number $\delta$ such that there are no points of the locus in the punctured disk
$$0 <x^2 +y^2 < \delta^2.$$
1959 Putnam, A1
Let $n$ be a positive integer. Prove that $x^n -\frac{1}{x^{n}}$ is expressible as a polynomial in $x-\frac{1}{x}$ with real coefficients if and only if $n$ is odd.