# Morning Session ## Problem 1 If $a_0, a_1, \ldots, a_n$ are real numbers satisfying $ \frac{a_0}{1}+\frac{a_1}{2}+\cdots+\frac{a_n}{n+1}=0, $ show that the equation $a_0+a_1 x+a_2 x^2+\cdots+a_n x^n=0$ has at least one real root. ## Problem 2 Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. ## Problem 3 Real numbers are chosen at random from the interval $(0 \leq x \leq 1)$. If after choosing the $n$th number the sum of the numbers so chosen first exceeds 1 , show that the expected or average value for $n$ is $e$. ## Problem 4 If $a_1, a_2, \ldots, a_n$ are complex numbers such that $\left|a_1\right|=\left|a_2\right|=\ldots=\left|a_n\right|=r \neq 0,$ and if $_n T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that $\left|\frac{_n T_s}{_n T_{n-s}}\right|=r^{2 s-n}$ whenever the denominator of the left-hand side is different from zero. ## Problem 5 Show that the integral equation $f(x, y)=1+\int_0^x \int_0^y f(u, v) d u d v$ has at most one solution continuous for $0 \leq x \leq 1,0 \leq y \leq 1$. ## Problem 6 What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order that one may withdraw 1 dollar at the end of the first year, 4 dollars at the end of the second year, $\ldots, n^2$ dollars at the end of the $n$th year, in perpetuity? ## Problem 7 Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others. # Afternoon Session ## Problem 1 > [!i] Given line segments $A, B, C, D$, with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible. > [!ii] Given any acute-angled triangle $A B C$ and one altitude $A H$, select any point $D$ on $A H$, then draw $B D$ and extend until it intersects $A C$ in $E$, and draw $C D$ and extend until it intersects $A B$ in $F$. Prove angle $A H E=$ angle $A H F$. ## Problem 2 Prove that the product of four consecutive positive integers cannot be a perfect square or cube. ## Problem 3 In a round-robin tournament with $n$ players (each pair of players plays one game) in which there are no draws, the numbers of wins scored by the players are $s_1, s_2, \ldots, s_n$. Prove that a necessary and sufficient condition for the existence of 3 players, $A, B, C$, such that $A$ beat $B, B$ beat $C$, and $C$ beat $A$ is $s_1{ }^2+s_2{ }^2+\cdots+s_n{ }^2<(n-1)(n)(2 n-1)^{\prime} 6 .$ ## Problem 4 What is the average straight line distance between two points on a sphere of radius 1? ## Problem 5 Given an infinite number of points in a plane, prove that if all the distances determined between them are integers then the points are all in a straight line. ## Problem 6 A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x$ $=f(t)$ gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $y=-g f(t) \int \frac{d t}{f^{\prime}(t)}+g \int \frac{f(t)}{f^{\prime}(t)} d t+A f(t)+B$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity. ## Problem 7 Prove that if $f(x)$ is continuous for $a \leq x \leq b$ and $\int_a^b x^n f(x) d x=0$ for $n=0,1,2, \ldots$ then $f(x)$ is identically zero on $a \leq x \leq b$.