# Morning Session ## Problem 1 Let $f(x)=\sum_{i=0}^{i \Sigma_n} a_i x^{n-1}$ be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n$, and $f(1)$ are odd, prove that $f(x)=0$ has no rational roots. ## Problem 2 Show that the equation $\left(9-x^2\right)\left(\frac{d y}{d x}\right)^2=\left(9-y^2\right)$ characterizes a family of conics touching the four sides of a fixed square. ## Problem 3 Develop necessary and sufficient conditions which ensure that $r_1, r_2, r_3$ and $r_1^2, r_2^2, r_3^2$ are simultaneously roots of the equation $x^3+a x^2+b x+c$ $=0$. ## Problem 4 The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances? ## Problem 5 Let $a_j(j=1,2, \ldots, n)$ be entirely arbitrary numbers except that no one is equal to unity. Prove $a_1+\sum_{i=2}^n a_i \prod_1^{i-1}\left(1-a_j\right)=1-\prod_{11}^n\left(1-a_j\right).$ ## Problem 6 A man has a rectangular block of wood $m$ by $n$ by $r$ inches $(m, n$, and $r$ are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.) ## Problem 7 Directed lines are drawn from the center of a circle, making angles of 0 , $\pm 1, \pm 2, \pm 3, \ldots$ (measured in radians from a prime direction). If these lines meet the circle in points $P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots$, show that there is no interval on the circumference of the circle which does not contain some $P_{s 1}$. (You may assume that $\pi$ is irrational.) # Afternoon Session ## Problem 1 A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2=b^2+c^2-2 b c \cos A$, to find the third side $a$. He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c ;$ subtracts the sum of the logarithms of 2, b, c, and $\cos A$; divides the result by 2 ; and takes the antilogarithm. Although his method may be open to suspicion, his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result? ## Problem 2 Find the surface generated by the solutions of $\frac{d x}{y z}=\frac{d y}{z x}=\frac{d z}{x y},$ which intersects the circle $y^2+z^2=1, x=0$. ## Problem 3 Develop necessary and sufficient conditions that the equation $\left|\begin{array}{ccc}0 & a_1-x & a_2-x \\-a_1-x & 0 & a_3-x \\-a_2-x & -a_3-x & 0\end{array}\right|=0 \quad\left(a_1 \neq 0\right)$ shall have a multiple root. ## Problem 4 A homogeneous solid body is made by joining a base of a circular cylinder of height $h$ and radius $r$, and the base of a hemisphere of radius $r$. This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if $r$ is large as compared to $h$, the equilibrium will be stable; but if $r$ is small as compared to $h$, the equilibrium will be unstable. What is the critical value of the ratio $r / h$ which enables the body to rest in neutral equilibrium in any position?