Putnam 1951

1. Show that the determinant: $$\left|\begin{array}{rrrr} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{array}\right|$$ is non-negative, if its elements $a, b, c$, etc., are real.

2. In the plane, what is the locus of points the sum of the squares of whose distances from $n$ fixed points is a constant? What restrictions, stated in geometric terms, must be put on the constant so that the locus is non-null?

\begin{array}{l} \text { 3. Find the sum to infinity of the series: }\ 1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots+\frac{(-1)^{n+1}}{3 n-2}+\cdots . (\text { page } 319) \end{array}

\begin{array}{l} \text { 4. Trace the curve whose equation is: }\ y^{4}-x^{4}-96 y^{2}+100 x^{2}=0 \end{array}

5. Consider in the plane the network of points having integral coordinates. For lines having rational slope show that: (i) the line passes through no points of the network or through infinitely many; (ii) there exists for each line a positive number $d$ having the property that no point of the network, except such as may be on the line, is closer to the line than the distance $d$.

6. Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.

1. Find the condition that the functions $M(x, y)$ and $N(x, y)$ must satisfy in order that the differential equation $M d x+N d y=0$ shall have an integrating factor of the form $f(x y)$. You may assume that $M$ and $N$ have continuous partial derivatives of all orders.

2. Two functions of $x$ are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.

\begin{array}{l} \text { 3. Show that if } x \text { is positive, then }\ \log _{e}(1+1 / x)>1^{\prime}(1+x) \end{array}

4. Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.

5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.

6. Assuming that all the roots of the cubic equation $x^{3}+a x^{2}+b x+c=$ 0 are real, show that the difference between the greatest and the least roots is not less than $\left(a^{2}-3 b\right)^{12}$ or greater than $2\left(a^{2}-3 b\right)^{12 / 3^{12}}$.