# Morning Session ## Problem 1 Prove that the polynomial $(a-x)^6-3 a(a-x)^5+\frac{5}{2} a^2(a-x)^4-\frac{1}{2} a^4(a-x)^2$ takes only negative values for $0<x<a$. ## Problem 2 Find the $n$th derivative with respect to $x$ of $\int_0^x\left[1+\frac{(x-t)}{1 !}+\frac{(x-t)^2}{2 !}+\cdots+\frac{(x-t)^{n-1}}{(n-1) !}\right] e^{n t} d t$ ## Problem 3 A circle of radius $a$ rolls in its plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter a, likewise rolling in its plane along the $\boldsymbol{x}$-axis. ## Problem 4 Let the roots $a, b, c$ of $f(x) \equiv x^3+p x^2+q x+r=0$ be real, and let $a \leq b \leq c$. Prove that, if the interval $(b, c)$ is divided into six equal parts, a root of $f^{\prime}(x)=0$ will lie in the fourth part counting from the end $b$. What will be the form of $f(x)$ if the root in question of $f^{\prime}(x)=0$ falls at either end of the fourth part? ## Problem 5 Show that the line which moves parallel to the plane $y=z$ and which intersects the two parabolas $y^2=2 x, z=0$ and $z^2=3 x, y=0$ sweeps out the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$ ## Problem 6 If the $x$-coordinate $\bar{x}$ of the center of mass of the area lying between the $x$-axis and the curve $y=f(x),(f(x)>0)$, and between the lines $x=0$ and $x$ $=a$ is given by $\bar{x}=g(a),$ show that $f(x)=A \frac{g^{\prime}(x)}{[x-g(x)]^2} e^{\int d x /(x-g(x))}$ where $A$ is a positive constant. ## Problem 7 Take either (i) or (ii). > [!i] Prove that > $ > \begin{array}{ccc} > \left|\begin{array}{ccc} > 1+a^2-b^2-c^2 & 2(a b+c) & 2(c a-b) \\ > 2(a b-c) & 1+b^2-c^2-a^2 & 2(b c+a) \\ > 2(c a+b) & 2(b c-a) & 1+c^2-a^2-b^2 > \end{array}\right| \\ > =\left(1+a^2+b^2+c^2\right)^3 > \end{array} > $ > [!ii] A semi-ellipsoid of revolution is formed by revolving about the $x$-axis the area lying within the first quadrant of the ellipse > $ > \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 > $ > Show that this semi-ellipsoid will balance in stable equilibrium, with its vertex resting on a horizontal plane, when and only when > $ > b \sqrt{8} \geq a \sqrt{5} . > $ # Afternoon Session ## Problem 8 A particle $(x, y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that $y\left(x^2+y^2+1\right) d x=x\left(x^2+y^2-1\right) d y,$ and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center. ## Problem 9 Evaluate the following limits: $ \begin{aligned} & \lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}}\right) \\ & \lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right) \\ & \lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}}\right) \end{aligned} $ ## Problem 10 Find the differential equation satisfied by the product $z$ of any two linearly independent integrals of the equation $ y^{\prime \prime}+y^{\prime} P(x)+y Q(x)=0 . $ ## Problem 11 Two perpendicular diameters of the ellipse $ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ are given, and the two diameters conjugate to them are constructed. Show that the rectangular hyperbola pàssing through the ends of these conjugate diameters passes through the foci of the ellipse. ## Problem 12 A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^2>g$. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is $ \frac{\left(a \omega+g \omega^{-1}\right)^2}{2 g} $ ## Problem 13 Assuming that $f(x)$ is continuous in the interval $(0,1)$, prove that $ \int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x) f(y) f(z) d x d y d z=\frac{1}{3 !}\left(\int_{t=0}^{t=1} f(t) d t\right)^3 . $ ## Problem 14 Take either `(i)` or `(ii)`. > [!i] Show that any solution $f(t)$ of the functional equation > $f(x+y) f(x-y)=f(x) f(x)+f(y) f(y)-1, \quad(x, y, \text { real })$ is such that $f^{\prime \prime}(t)=\pm m^2 f(t), \quad(m \text { constant and } \geq 0),$ assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $\pm \cos m t, \quad \pm \cosh m t .$ > [!ii] With $n$ constant values $a_1, a_2, \ldots, a_n$, supposed all different, let $n$ constant values $b_1, b_2, \ldots, b_n$ be associated, and let a polynomial $P(x)$ be defined by the identity in $x$ > $\left|\begin{array}{cccccc}1 & x & x^2 & \cdots & x^{n-1} & P(x) \\1 & a_1 & a_1^2 & \cdots & a_{1^{n-1}} & b_1 \\1 & a_2 & a_2^2 & \cdots & a_{2^{n-1}} & b_2 \\\cdots \cdots & \ldots & \ldots & \ldots & \ldots & \cdots \\1 & a_n & a_n^2 & \cdots & a_n^{n-1} & b_n\end{array}\right| \equiv 0$ > Given a polynomial $\phi(t)$, let a polynomial $Q(x)$ be defined by the identity in $x$ obtained on replacing $P(x), b_1, b_2, \ldots, b_n$ of the identity above by $Q(x)$, $\phi\left(b_1\right), \phi\left(b_2\right), \ldots, \phi\left(b_n\right)$. Prove that the remainder obtained on dividing $\phi(P(x))$ by $\left(x-a_1\right)\left(x-a_2\right) \cdots\left(x-a_n\right)$ is $Q(x)$.