# Morning Session ## Problem 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. ## Problem 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? ## Problem 3 Find the sum to infinity of the series:$1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots+\frac{(-1)^{n+1}}{3 n-2}+\cdots$ ## Problem 4 Trace the curve whose equation is:$y^4-x^4-96 y^2+100 x^2=0 .$ ## Problem 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 $d$. ## Problem 6 Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area. ## Problem 7 Show that if the series $a_1+a_2+a_3+\cdots+a_n+\cdots$ converges, then the series $a_1+a_2 2+a_3 3+\cdots+a_n^{\prime} n+\cdots$ converges also. # Afternoon Session ## Problem 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. ## Problem 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. ## Problem 3 Show that if $x$ is positive, then $\log _e(1+1 / x)>1^{\prime}(1+x) . $ ## Problem 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. ## Problem 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. ## Problem 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 $2\left(a^2-3 b\right)^{12 / 3^{12}}$. ## Problem 7 Find the volume of the four-dimensional hypersphere $x^2+y^2+z^2+t^2=r^2$, and also the hypervolume of its interior $x^2+y^2+z^2+t^2<r^2.$