# Morning Session ## Problem 1 Suppose that the function $f(x)=a x^2+b x+c$, where $a, b, c$ are real constants, satisfies the condition $|f(x)| \leq 1$ for $|x| \leq 1$. Prove that $\left|f^{\prime}(x)\right|$ $\leq 4$ for $|x| \leq 1$. ## Problem 2 If $a(x), b(x), c(x)$, and $d(x)$ are polynomials in $x$, show that $\int_1^x a(x) c(x) d x \cdot \int_1^x b(x) d(x) d x-\int_1^x a(x) d(x) d x \cdot \int_1^x b(x) c(x) d x$ is divisible by $(x-1)^4$. ## Problem 3 A projectile in flight is observed simultaneously from four radar stations which are situated at the corners of a square of side $b$. The distances of the projectile from the four stations, taken in order around the square, are found to be $R_1, R_2, R_3, R_4$. Show that $R_1{ }^2+R_3{ }^2=R_2{ }^2+R_4{ }^2 .$ Show also that the height $h$ of the projectile above the ground is given by $ \begin{aligned} h^2= & -\frac{1}{2} b^2+\frac{1}{4}\left({R_1}^2+{R_2}^2+{R_3}^2+{R_4}^2\right) \\ & -\frac{1}{8 b^2}\left({R_1}^4+{R_2}^4+{R_3}^4+{R_4}^4-2 {R_1}^2 {R_3}^2-2 {R_2}^2 {R_4}^2\right) \end{aligned} $ ## Problem 4 Let $g(x)$ be a function that has a continuous first derivative $g^{\prime}(x)$ for all values of $x$. Suppose that the following conditions hold for every $x:(i) g(0)=$ 0 ; (ii) $\left|g^{\prime}(x)\right| \leq|g(x)|$. Prove that $g(x)$ vanishes identically. ## Problem 5 Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ ## Problem 6 A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of this function is not known. A motion is observed, and the distance $x$ covered in time $t$ is found to be connected with $t$ by the formula $x=a t+b t^2+c t^3$, where $a, b, c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment. # Afternoon Session ## Problem 1 Let $K$ denote the circumference of a circular disc of radius one, and let $k$ denote a circular arc that joins two points $a, b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disc into two parts of equal area. Prove that the length of $k$ exceeds 2. ## Problem 2 Let $A, B$ be variable points on a parabola $P$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A, B$ and the vertex of $P$ is a parabola $P_1$. Apply the same process to $P_1$, obtaining a parabola $P_2$, and repeat the process, obtaining altogether the sequence of parabolas $P, P_1, P_2, \ldots, P_n$. If the equation of $P$ is $y^2=m x$, find the equation of $P_n$. ## Problem 3 In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r$. Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center. (Assume that the magnitude of the force of attraction at a point $P$ due to a thin spherical shell is zero if $P$ is inside the shell, and is $m / r^2$ if $P$ is outside the shell, $m$ being the mass of the shell, and $r$ the distance of $P$ from the center.) ## Problem 4 For each positive integer $n$, put $p_n=(1+1 / n)^n, P_n=(1+1 / n)^{n+1}, h_n=\frac{2 p_n P_n}{p_n+P_n}$ Prove that $h_1<h_2<\cdots<h_n<\cdots$. ## Problem 5 Show that the integer next above $(\sqrt{3}+1)^{2 n}$ is divisible by $2^{n+1}$. ## Problem 6 A particle moves on a circle with center $O$, starting from rest at a point $P$ and coming to rest again at a point $Q$, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $Q$ and that, at some point $R$ between $P$ and $Q$, the acceleration vector points in along the radius $R O$.