# Morning Session ## Problem 1 Prove that if $f(x)$ is a polynomial with integral coefficients, and there exists an integer $k$ such that none of the integers $f(1), f(2), \ldots, f(k)$ is divisible by $k$, then $f(x)$ has no integral root. ## Problem 2 Let $A$ and $B$ be two fixed points on the curve $y=f(x)$, where $f(x)$ is continuous and has a continuous derivative, and the $\operatorname{arc} A B$ is concave to the chord $A B$. If $P$ is a point of the arc $A B$ for which $A P+P B$ is a maximum, prove that $P A$ and $P B$ are equally inclined to the tangent to the curve $y=$ $f(x)$ at the point $P$. ## Problem 3 Find $f(x)$ such that $ \int[f(x)]^n d x=\left[\int f(x) d x\right]^n, $ when constants of integration are suitably chosen. ## Problem 4 The parabola $y^2=-4 p x$ rolls without slipping around the parabola $y^2=4 p x$. Find the equation of the locus of the vertex of the rolling parabola. ## Problem 5 Prove that the simultaneous equations $ x^4-x^2=y^4-y^2=z^4-z^2 $ are satisfied by the points of four straight lines and six ellipses, and by no other points. ## Problem 6 $f(x)$ is a polynomial of degree $n$, such that a power of $f(x)$ is divisible by a power of its derivative $f^{\prime}(x)$; i.e., $[f(x)]^p$ is divisible by $\left[f^{\prime}(x)\right]^q ; p, q$, positive integers. Prove that $f(x)$ is divisible by $f^{\prime}(x)$ and that $f(x)$ has a single root of multiplicity $\boldsymbol{n}$. ## Problem 7 If $u_1^2+u_2{ }^2+\cdots$ and $v_1{ }^2+v_2{ }^2+\cdots$ are convergent series of real constants, prove that $\left(u_1-v_1\right)^p+\left(u_2-v_2\right)^p+\cdots, p \text { an integer } \geq 2$ is convergent. ## Problem 8 A triangle is bounded by the lines $ \begin{aligned} A_1 x+B_1 y+C_1 & =0, \quad A_2 x+B_2 y+C_2=0 \\ A_3 x & +B_3 y+C_3=0 \end{aligned} $ Show that the area, disregarding sign, is $\frac{\left|\begin{array}{lll}A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3\end{array}\right|^2}{2\left|\begin{array}{cc}A_2 & B_2 \\ A_3 & B_3\end{array}\right| \cdot\left|\begin{array}{ll}A_3 & B_3 \\ A_1 & B_1\end{array}\right| \cdot\left|\begin{array}{ll}A_1 & B_1 \\ A_2 & B_2\end{array}\right|}$ # Afternoon Session ## Problem 9 A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the length of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $ \sin \alpha \log (\sec \alpha+\tan \alpha)=1 . $ ## Problem 10 A cylindrical hole of radius $r$ is bored through a cylinder of radius $R$ $(r \leq R)$ so that the axes intersect at right angles. (i) Show that the area of the larger cylinder which is inside the smaller can be expressed in the form $ S=8 r^2 \int_0^1 \frac{1-v^2}{\sqrt{\left(1-v^2\right)\left(1-m^2 v^2\right)}} d v \quad \text { where } m=\frac{r}{R} . $ (ii) If $ K=\int_0^1 \frac{d v}{\sqrt{\left(1-v^2\right)\left(1-m^2 v^2\right)}} \text { and } E=\int_0^1 \sqrt{\frac{1-m^2 v^2}{1-v^2}} d v $ show that $ S=8\left[R^2 E-\left(R^2-r^2\right) K\right] . $ ## Problem 11 From any point $(a, b)$ in the Cartesian plane, show that `(i)` three normals, real or imaginary, can be drawn to the parabola $y^2=4 p x$; `(ii)` these are real and distinct if $4(2 p-a)^3+27 p b^2<0$; (iii) two of them coincide if $(a, b)$ lies on the curve $27 p y^2=4(x-2 p)^3$; (iv) all three coincide only if $(a, b)$ is the point $(2 p, 0)$. ## Problem 12 Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface $a x^2+b y^2+c z^2=1 \quad(a b c \neq 0)\tag{1}$ is the sphere $x^2+y^2+z^2=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\tag{2}$ ## Problem 13 Determine all rational values for which $a, b, c$ are the roots of $x^3+a x^2+b x+c=0$ ## Problem 14 Prove that $ \left(\begin{array}{lllll} a_1{ }^2+k & a_1 a_2 & a_1 a_3 & \ldots & a_1 a_n \\ a_2 a_1 & a_2{ }^2+k & a_2 a_3 & \ldots & a_2 a_n \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_n a_1 & a_n a_2 & a_n a_3 & \ldots & a_n{ }^2+k \end{array}\right) $ is divisible by $k^{n-1}$ and find its other factor. ## Problem 15 Which is greater $\left(\sqrt{n\phantom{+\!\!\!\!}}\right)^{\sqrt{n+1}}\text{\quad or\quad}\left(\sqrt{n+1}\right)^{\sqrt{n}}$ where $n>8 ?$