Found problems: 966
1998 Putnam, 2
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists.
2013 Putnam, 3
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
1985 Putnam, A2
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.
1972 Putnam, A3
A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists.
Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.
1985 Putnam, A5
Let $I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .$ For which integers $m, 1 \leq m \leq 10$ is $I_{m} \neq 0 ?$
1941 Putnam, B4
Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$.
Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.
1957 Putnam, B4
Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$'s and $2$'s. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$.
2009 Putnam, B1
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\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$
2000 Putnam, 1
Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$?
2022 Putnam, A1
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.
1955 Putnam, A2
$A_1 ~A_2~ \ldots ~A_n$ is a regular polygon inscribed in a circle of radius $r$ and center $O.$ $P$ is a point on line $OA_1$ extended beyond $A_1.$ Show that \[ \prod^n_{i=1} ~ \overline{PA}_{~i} = \overline{OP}^{~n} - r^n. \]
1980 Putnam, B3
For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$
2006 Romania Team Selection Test, 4
Let $x_i$, $1\leq i\leq n$ be real numbers. Prove that \[ \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. \]
[i]Discrete version by Dan Schwarz of a Putnam problem[/i]
2003 Putnam, 5
A Dyck $n$-path is a lattice path of $n$ upsteps $(1, 1)$ and $n$ downsteps $(1, -1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. For example, the Dyck $5$-path illustrated has two returns, of length $3$ and $1$ respectively. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n - 1)$ paths.
\[\begin{picture}(165,70)
\put(-5,0){O}
\put(0,10){\line(1,0){150}}
\put(0,10){\line(1,1){30}}
\put(30,40){\line(1,-1){15}}
\put(45,25){\line(1,1){30}}
\put(75,55){\line(1,-1){45}}
\put(120,10){\line(1,1){15}}
\put(135,25){\line(1,-1){15}}
\put(0,10){\circle{1}}\put(0,10){\circle{2}}\put(0,10){\circle{3}}\put(0,10){\circle{4}}
\put(15,25){\circle{1}}\put(15,25){\circle{2}}\put(15,25){\circle{3}}\put(15,25){\circle{4}}
\put(30,40){\circle{1}}\put(30,40){\circle{2}}\put(30,40){\circle{3}}\put(30,40){\circle{4}}
\put(45,25){\circle{1}}\put(45,25){\circle{2}}\put(45,25){\circle{3}}\put(45,25){\circle{4}}
\put(60,40){\circle{1}}\put(60,40){\circle{2}}\put(60,40){\circle{3}}\put(60,40){\circle{4}}
\put(75,55){\circle{1}}\put(75,55){\circle{2}}\put(75,55){\circle{3}}\put(75,55){\circle{4}}
\put(90,40){\circle{1}}\put(90,40){\circle{2}}\put(90,40){\circle{3}}\put(90,40){\circle{4}}
\put(105,25){\circle{1}}\put(105,25){\circle{2}}\put(105,25){\circle{3}}\put(105,25){\circle{4}}
\put(120,10){\circle{1}}\put(120,10){\circle{2}}\put(120,10){\circle{3}}\put(120,10){\circle{4}}
\put(135,25){\circle{1}}\put(135,25){\circle{2}}\put(135,25){\circle{3}}\put(135,25){\circle{4}}
\put(150,10){\circle{1}}\put(150,10){\circle{2}}\put(150,10){\circle{3}}\put(150,10){\circle{4}}
\end{picture}\]
1953 Putnam, B2
Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$
1980 Putnam, A4
a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that
$$ |a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$
b) Let $ a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that
$$ |a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$
1946 Putnam, A3
A projectile in flight is observed simultaneously from four radio stations which are situated at the corners of a square of side $b$. The distances of the projectile from the four stations, taken in order around the square, are found to be $R_1 , R_2 , R_3 $ and $R_4$. Show that
$$R_{1}^{2}+ R_{3}^{2}= R_{2}^{2}+ R_{4}^{2}.$$
Show also that the height $h$ of the projectile above the ground is given by
$$h^{2}=- \frac{1}{2} b^2 +\frac{1}{4}(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}) -\frac{1}{8 b^{2}}(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}- 2 R_{1}^{2}R_{3}^{2} -2 R_{2}^{2} R_{4}^{2}).$$
1961 Putnam, B3
Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.
1994 Putnam, 4
Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.
1973 Putnam, B2
Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.
2007 Putnam, 1
Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.
2009 Putnam, A1
Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?
1967 Putnam, A4
Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds:
$$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$
1995 Putnam, 1
Let $S$ be a set of real numbers which is closed under multiplication (that is $a,b\in S\implies ab\in S$). Let $T,U\subset S$ such that $T\cap U=\emptyset, T\cup U=S$. Given that for any three elements $a,b,c$ in $T$, not necessarily distinct, we have $abc\in T$ and also if $a,b,c\in U$, not necessarily distinct then $abc\in U$. Show at least one of $T$ and $U$ is closed under multiplication.
2019 Putnam, A2
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$. Find $\alpha$.