# Morning Session
## Problem 1
If $\left\{a_n\right\}$ is a sequence of numbers such that for $n \geq 1$ $\left(2-a_n\right) a_{n+1}=1,$ prove that $\lim a_n$, as $n \rightarrow \infty$, exists and is equal to one.
## Problem 2
A real valued continuous function satisfies for all real $x$ and $y$ the functional equation $f\left(\sqrt{x^2+y^2}\right)=f(x) f(y)$ Prove that $f(x)=[f(1)]^{x^2}$
## Problem 3
Given this figure![[1947_Putnam_Problem3.svg]] and any two points $Q_1, Q_2$ in the plane not lying on any of the segments $s_1, s_2, \ldots, s_6$, show that there does not exist a polygonal line $P$ joining $Q_1$ and $Q_2$ such that:
> [!i] $P$ crosses each $s_i, i=1,2, \ldots, 6$, exactly once;
> [!ii] $P$ does not intersect itself;
> [!iii] $P$ does not pass through any vertex $V_1, V_2, V_3, V_4$.
## Problem 4
A coast artillery gun can fire at any angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant $\left(=v_0\right)$, determine the set $H$ of points in the plane and above the horizontal which can be hit.
## Problem 5
$a_1, b_1, c_1$ are positive numbers whose sum is 1 , and for $n=1,2, \ldots$ we define $a_{n+1}=a_n^2+2 b_n c_n, b_{n+1}=b_n^2+2 c_n a_n, c_{n+1}=c_n^2+2 a_n b_n .$ Show that $a_n, b_n, c_n$ approach limits as $n \rightarrow \infty$ and find these limits.
## Problem 6
A three-by-three matrix has determinant zero, and has the further property that the cofactor of any element is equal to the square of that element. (The cofactor of $a_{i j}$ is $(-1)^{i+j}$ multiplied by the determinant obtained by striking out the $i$-th row and $j$-th column.) Show that every element in the matrix is zero.
# Afternoon Session
## Problem 7
Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $f^{\prime}(x)=\frac{1}{x^2+f^2(x)}.$ Prove that $\lim _{x \rightarrow \infty} f(x)$ exists and is less than $1+\pi / 4$.
## Problem 8
Let $f(x)$ be a differentiable function defined in the closed interval $(0,1)$ and such that $\left|f^{\prime}(x)\right| \leq M, \quad 0<x<1$ Prove that$\left|\int_0^1 f(x) d x-\frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right)\right| \leq \frac{M}{n} .$
## Problem 9
Let $x, y$ be Cartesian coordinates in the plane. $I$ denotes the line segment $1 \leq x \leq 3, y=1$. For every point $P$ on $I$, let $P^*$ denote that point that lies on the segment joining the origin to $P$ and such that distance $P P *$ is equal to $1 / 100$. As $P$ describes $I$, the corresponding point $P^*$ describes a certain curve $C^*$. Let $l(I), l\left(C^*\right)$ be the lengths of $I$ and $C^*$ respectively. Which one of $l(I), l\left(C^*\right)$ is greater? Prove your answer.
## Problem 10
Given $P(z)=z^2+a z+b$, a quadratic polynomial of the complex variable $z$ with complex coefficients $a, b$. Suppose that $|P(z)|=1$ for every $z$ such that $|z|=1$. Prove that $a=b=0$.
## Problem 11
$a, b, c, d$ are distinct integers such that $(x-a)(x-b)(x-c)(x-d)-4=0$ has an integral root $r$. Show that $4 r=a+b+c+d$
## Problem 12
$C$ is a fixed point on $O Z$ and $U, V$ are variable points on $O X, O Y$ respectively, where $O X, O Y, O Z$ are mutually orthogonal lines. Find the locus of a point $P$ such that $P U, P V, P C$ are mutually orthogonal.
%% Morning Problem 3 asy
size(8cm);
pair V1,V2,V3,V4;
V1=(sqrt(3)/2,-0.5);
V2=(0,1);
V3=(-sqrt(3)/2,-0.5);
V4=(0.1,0.1);
draw(V1--V2--V3--cycle);
draw(V1--V4);
draw(V2--V4);
draw(V3--V4);
label("$V_1
quot;,V1,ESE);
label("$V_2quot;,V2,N);
label("$V_3quot;,V3,WSW);
label("$V_4quot;,V4-(0,0.05),S);
label("$s_1quot;,midpoint(V2--V3),NW);
label("$s_2quot;,midpoint(V1--V3),S);
label("$s_3quot;,midpoint(V2--V1),NE);
label("$s_4quot;,midpoint(V1--V4),WSW);
label("$s_5quot;,midpoint(V2--V4),SW);
label("$s_6quot;,midpoint(V3--V4),SE);
%%