Found problems: 966
2002 Greece Junior Math Olympiad, 4
Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$
1947 Putnam, B4
Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$
2016 Putnam, A6
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1],$
\[\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.\]
2012 Putnam, 4
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)
1962 Putnam, A5
Evaluate
$$ \sum_{k=0}^{n} \binom{n}{k}k^{2}.$$
1994 Putnam, 2
For which real numbers $c$ is there a straight line that intersects the curve
\[ y = x^4 + 9x^3 + cx^2 + 9x + 4\]
in four distinct points?
1967 Putnam, B2
Let $0\leq p,r\leq 1$ and consider the identities
$$a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.$$
Show that
$$ a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.$$
2001 Putnam, 2
Find all pairs of real numbers $(x,y)$ satisfying the system of equations:
\begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}
2007 Putnam, 4
A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
1958 November Putnam, A4
In assigning dormitory rooms, a college gives preference to pairs of students in this order:
$$AA,\, AB ,\, AC, \, BB , \, BC ,\, AD , \, CC, \, BD, \, CD, \, DD$$
in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A,B,C$ and $D$ so that the set of numbers $A+A, A+B, A+C, B+B, \ldots $ corresponding to the order above will be in descending order. Find the general solution and the solution in least positive integers.
Putnam 1939, B2
Evaluate $\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx$ and $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.$
2001 IMC, 3
Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.
2019 Putnam, A4
Let $f$ be a continuous real-valued function on $\mathbb R^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero. Must $f(x,y,z)$ be identically zero?
Putnam 1938, A3
A particle moves in the Euclidean plane. At time $t$ (taking all real values) its coordinates are $x = t^3 - t$ and $y = t^4 + t.$ Show that its velocity has a maximum at $t = 0,$ and that its path has an inflection at $t = 0.$
1961 Putnam, B2
Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$
1955 Putnam, A6
Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.
2014 Putnam, 4
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
1967 Putnam, A5
Show that in a convex region in the plane whose boundary contains at most a finite number of straight line segments and whose area is greater than $\frac{\pi}{4}$ there is at least one pair of points a unit distance apart.
1941 Putnam, A5
The line $L$ is parallel to the plane $y=z$ and meets the parabola $2x=y^2 ,z=0$ and the parabola $3x=z^2, y=0$. Prove that if $L$ moves freely subject to these constraints then it generates the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$.
2010 Putnam, A1
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
1978 Putnam, A6
Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.
2008 Putnam, B3
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?
2004 Putnam, B1
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers
$c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$
are all integers.
1989 Putnam, B2
Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group?
I haven't had much group theory at this point...
2007 Putnam, 3
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.