Found problems: 966
1941 Putnam, B1
A particle $(x,y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that
$$y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy,$$
and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center.
2023 Putnam, A1
For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$.
1973 Putnam, B4
(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that
$$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$
(b) Show an example in which equality occurs.
1949 Putnam, B6
Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows:
Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular.
Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)
2017 Putnam, B5
A line in the plane of a triangle $T$ is called an [i]equalizer[/i] if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c,$ with $a$ as small as possible, such that there exists a triangle with side lengths $a,b,c$ that has exactly two distinct equalizers.
1952 Putnam, B5
If the terms of a sequence $a_{1}, a_{2}, \ldots$ are monotonic, and if $\sum_{n=1}^{\infty} a_n$ converges, show that $\sum_{n=1}^{\infty} n(a_{n} -a_{n+1 })$ converges.
1963 Putnam, B4
Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle?
1985 Putnam, B5
Evaluate $\textstyle\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t.$ You may assume that $\textstyle\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}.$
2008 Putnam, A1
Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$
1940 Putnam, A2
Let $A,B$ be two fixed points on the curve $y=f(x)$, $f$ is continuous with continuous derivative and the arc $\widehat{AB}$ is concave to the chord $AB$. If $P$ is a point on the arc $\widehat{AB}$ for which $AP+PB$ is maximal, prove that $PA$ and $PB$ are equally inclined to the tangent to the curve $y=f(x)$ at $P$.
2013 Putnam, 5
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
2000 Putnam, 2
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: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]
1972 Putnam, B3
A group $G$ has elements $g,h$ satisfying $ghg=hg^{2}h, g^{3}=1$ and $h^n=1$ for some odd integer $n$. Prove that $h=e$, where $e$ is the identity element.
2022 Putnam, B3
Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate the recoloring process, will we always end up with all the numbers red after a finite number of steps?
1970 Putnam, A4
Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that
$$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$
2004 Putnam, A2
For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?
1952 Putnam, B7
Given any real number $N_0,$ if $N_{j+1}= \cos N_j ,$ prove that $\lim_{j\to \infty} N_j$ exists and is independent of $N_0.$
1970 Putnam, B4
An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute, along a straight road. If a governor prevents the speed of the car from exceeding $90$ miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least $6.6$ ft/sec.
2022 Putnam, A6
Let $n$ be a positive integer. Determine, in terms of $n,$ the largest integer $m$ with the following property: There exist real numbers $x_1,\ldots, x_{2n}$ with $-1<x_1<x_2<\ldots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $$[x_1^{2k-1},x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \ldots, [x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$$ is equal to 1 for all integers $k$ with $1\leq k \leq m.$
1953 Putnam, B7
Let $w\in (0,1)$ be an irrational number. Prove that $w$ has a unique convergent expansion of the form
$$w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,$$
where $1\leq p_0 < p_1 < p_2 <\ldots $ are integers. If $w= \frac{1}{\sqrt{2}},$ find $p_0 , p_1 , p_2.$
2006 VTRMC, Problem 3
Hey,
This problem is from the VTRMC 2006.
3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6).
As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation.
Consider the following,
01 = 1 mod 10
01 = 1 mod 10
02 = 2 mod 10
03 = 3 mod 10
05 = 5 mod 10
08 = 6 mod 10
13 = 3 mod 10
21 = 1 mod 10
34 = 4 mod 10
55 = 5 mod 10
89 = 9 mod 10
Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships.
$ F(55) \equal{} F(05) \plus{} 5({2}^{2})$
This is pretty much as far as we got, any ideas?
2001 Putnam, 5
Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
2001 Putnam, 1
Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.
2006 Putnam, A4
Let $S=\{1,2\dots,n\}$ for some integer $n>1.$ Say a permutation $\pi$ of $S$ has a local maximum at $k\in S$ if
\[\begin{array}{ccc}\text{(i)}&\pi(k)>\pi(k+1)&\text{for }k=1\\ \text{(ii)}&\pi(k-1)<\pi(k)\text{ and }\pi(k)>\pi(k+1)&\text{for }1<k<n\\ \text{(iii)}&\pi(k-1)M\pi(k)&\text{for }k=n\end{array}\]
(For example, if $n=5$ and $\pi$ takes values at $1,2,3,4,5$ of $2,1,4,5,3,$ then $\pi$ has a local maximum of $2$ as $k=1,$ and a local maximum at $k-4.$)
What is the average number of local maxima of a permutation of $S,$ averaging over all permuatations of $S?$
1961 Putnam, B6
Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.