Putnam 1980

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$ depending only on $r$ and $s.$

a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$|a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$|a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$

Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $$\int_{0}^{x} P(t)\sin t dt =0, \;\;\; \int_{0}^{x} P(t) \cos t dt =0$$ has only finitely many solutions $x.$

Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1 .$ Determine the largest real number $u$ such that $$u \leq \int_{0}^{1} |f'(x) -f(x) | dx$$ for all $f$ in $C.$

For which real numbers $c$ is $$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$ for all real $x?$

Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six simultaneous conditions: $$x,y,z \geq 0, \; x+y+z\leq 11, \; 2x+4y+3z \leq 36, \; 2x+3z \leq 44.$$ a) Determine the number $V$ of vertices of $S.$ b) Determine the number $E$ of edges of $S.$ c) Sketch in the $bc$ -plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$

For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$

Let $A_1 , A_2 ,\ldots, A_{1066}$ be subsets of a finite set $X$ such that $|A_i | > \frac{1}{2} |X|$ for $1\leq i \leq 1066.$ Prove that there exist ten elements $x_1 ,x_2 ,\ldots , x_{10}$ of $X$ such that every $A_i$ contains at least one of $x_1 , x_2 ,\ldots, x_{10}.$