# Morning Session ## Problem 1 Prove that, for every positive integer $n$, $\sqrt{1}+\sqrt{2}+\cdots+\sqrt{n}$ is more than $\frac{2}{3} n \sqrt{n}$ and less than $\frac{4 n+3}{6} \sqrt{n}.$ ## Problem 2 Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color. ## Problem 3 If $x_1, x_2, x_3$ are real numbers and the sum of any two is greater than the third, show that $\frac{2}{3} \sum_{i=1}^3 x_i \sum_{i=1}^3 {x_i}^2>\sum_{i=1}^3 {x_i}^3+x_1 x_2 x_3 .$ ## Problem 4 From the identity $ \begin{aligned} \int_0^{\pi / 2} \log \sin 2 x d x & =\int_0^{\pi / 2} \log \sin x d x \\ & +\int_0^{\pi / 2} \log \cos x d x+\int_0^{\pi / 2} \log 2 d x, \end{aligned} $ deduce the value of $\int_0^{\pi / 2} \log \sin x d x .$ ## Problem 5 Let $P$ be a point from which three distinct normals can be drawn to a parabola. Show that the sum of the angles which these three normals make with the axis exceeds by a multiple of $\pi$ the angle which the line joining $P$ to the focus makes with the axis. ## Problem 6 Show that the sequence $\sqrt{7,} \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7+\sqrt{7}},} \sqrt{7-\sqrt{7+\sqrt{7-\sqrt{7}}}} \ldots$ converges, and evaluate the limit. ## Problem 7 Assuming that the roots of $x^3+p x^2+q x+r=0$ are all real and positive, find the relation between $p, q$, and $r$ which is a necessary and sufficient condition that the roots may be the cosines of the angles of a triangle. # Afternoon Session ## Problem 1 Is the infinite series $\sum_{n=1}^{\infty} \frac{1}{n^{(n+1) / n}}$ convergent? Prove your statement. ## Problem 2 Let $a_0, a_1, \ldots, a_n$ be real numbers and let $f(x)=a_0+a_1 x+\ldots+$ $a_n x^n$. Suppose that, for every integer $i, f(i)$ is an integer. Prove that $n ! \cdot a_k$ is an integer for each $k$. ## Problem 3 Solve the equations $\frac{d y}{d x}=z(y+z)^n \quad \frac{d z}{d x}=y(y+z)^n,$ given the initial conditions $y=1$ and $z=0$ when $x=0 . \quad$ ## Problem 4 Determine the equation of a surface in three dimensional cartesian space which has the following properties: `(a)` it passes through the point $(1,1$, 1); and `(b)` if the tangent plane be drawn at any point $P$, and $A, B$ and $C$ are the intersections of this plane with the $x, y$ and $z$ axes respectively, then $P$ is the orthocenter (intersection of the altitudes) of the triangle $A B C$. ## Problem 5 Show that the roots of $x^4+a x^3+b x^2+c x+d=0$, if suitably numbered, satisfy the relation $r_1 / r_2=r_3 / r_4$, provided $a^2 d=c^2 \neq 0$. ## Problem 6 $P$ and $Q$ are any points inside a circle $(C)$ with center $C$, such that $C P$ $=C Q$. Determine the location of a point $Z$ on $(C)$ such that $P Z+Q Z$ shall be a minimum. ## Problem 7 Let $w$ be an irrational number with $0<w<1$. Prove that $w$ has a unique convergent expansion of the form $w=\frac{1}{p_0}-\frac{1}{p_0 p_1}+\frac{1}{p_0 p_1 p_2}-\frac{1}{p_0 p_1 p_2 p_3}+\cdots,$ where $p_0, p_1, p_2, \ldots$ are integers and $1 \leq p_0<p_1<p_2<\cdots$. <br> If $w=$ $\frac{1}{2}$, find $p_0, p_1, p_2$.