Putnam 1953

Prove that for every positive integer $n$ $$\frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.$$

The complete graph with 6 points and 15 edges has each edge colored red or blue. Show that we can find 3 points such that the 3 edges joining them are the same color.

$a, b, c$ are real, and the sum of any two is greater than the third.

Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$ .

S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.

Show that the sequence $$\sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$ converges and evaluate the limit.

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

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 $k.$

Solve the equations $$\frac{dy}{dx}=z(y+z)^n, \; \frac{dz}{dx} = y(y+z)^n,$$ given the initial conditions $y=1$ and $z=0$ when $x=0.$

Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point $(1,1,1)$ and (b) if the tangent plane is drawn at any point $P$ and $X,Y, Z$ are the intersections of this plane with the $x, y$ and $z-$ axis respectively, then $P$ is the orthocenter of the triangle $XYZ.$

Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$ , if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$

Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.