Putnam 1978

Let $A$ be any set of $20$ distinct integers chosen from the arithmetic progression $1, 4, 7,\ldots,100$ . Prove that there must be two distinct integers in $A$ whose sum is $104$ .

Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$ . Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$ . Show that $$\text{det} \begin{pmatrix} p_1 & a& a & \cdots & a b & p_2 & a & \cdots & a b & b & p_3 & \cdots & a \vdots & \vdots & \vdots & \ddots & \vdots b & b& b &\cdots &p_n \end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$

Find the value of $k\ (0<k<5)$ such that $\int_0^{\infty} \frac{x^k}{2+4x+3x^2+5x^3+3x^4+4x^5+2x^6}\ dx$ is minimal.

A bypass operation on a set $S$ is a mapping $B: S\times S \rightarrow S$ with the property $B(B(w, x), B(y,z)) = B(w,z)$ for all $w, x, y, z \in S$ . (a) Prove that $B(a,b)=c$ implies $B(c,c)=c$ when $B$ is a bypass. (b) Prove that $B(a,b)=c$ implies $B(a,x)=B(c,x)$ for all $x\in S$ when $B$ is a bypass. (c) Construct a bypass operation $B$ on a finite set S with the following three properties [list=i] [] $B(x,x)=x$ for all $x\in S$ . [] There exist $d$ and $e$ in $S$ with $B(d,e)=d \ne e.$ [*] There exist $f$ and $g$ in $S$ with $B(f,g)\ne f.$ [/list]

Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set $$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$ Prove that $$\prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$

Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and the remaining four sides of length $2$ . Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.

Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$ as a rational number.

The sequence $(Q_{n}(x))$ of polynomials is defined by $$Q_{1}(x)=1+x , Q_{2}(x)=1+2x,$$ and for $m \geq 1$ by $$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$ $$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$ Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$

Prove that for every real number $N$ the equation $$x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$ has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3$ and $x_4$ are all larger than $N.$

Find the largest $a$ for which there exists a polynomial $$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$ with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$

Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ( $h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$ ) satisfy $0 \leq c_{hk} \leq 1.$ Prove that $$\left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$ where each summation is over all admissible ordered pairs $(h,k).$