Problems

1. Find the least value of $$\frac{x}{y}+\frac{3y}{z}+\frac{9z}{x}$$ over all positive real numbers $x,y,$and z.
2. For any positive constant $c$, find the maximum value of $xy(c-x-y)$ over all positive numbers $x$ and $y$.
3. If $a$ is any positive constant, find the minimum value of $$x^2 +\frac{a}{x}$$.
4. Find the maximum value of the product $xy(72-3x-4y)$ for positive $x$ and $y$.
5. Find the dimensions of the box of maximum volume that has one corner at the origin, three sides that contain that corner lying in the three coordinate planes and the opposite corner lying on the plane $2x+3y+4z =12$ in the first octant. (How to Ace the rest of Calculus)
6. Find the smallest value of 5x + 16/x + 21.
7. Find the maximum and minimum values, if any, of the function $f(x) =\sqrt{100+x^2}-x$ over the domain $x\ge0$.
8. Find the least value of the sum $x^2 +4x +4/x +1/x^2$ over positive real numbers $x$.
9. Find the least value of $$f(x) = \frac{(x+10)(x+2)}{x+1}$$.
10. Find the maximum value of $x^2y$ if $x$ and $y$ are restricted to positive real numbers satisfying $6x+5y=45$.
11. For any positive constant a, find the maximum of $$\frac{x}{x^2+a}$$ all positive $x$.
12. For any positive constant a, find the maximum of $$\frac{x^2}{x^3+a}$$ all positive $x$.
13. A manufacturer makes aluminum cups of volume 16 cubic inches in the form of right circular cylinders. Find the dimensions that use the least material. (OHS, Jan 2002)
14. Minimize the expression $6x +24/x^2$ over positive numbers $x$.
15. Find the maximum value of $$\frac{12(xy -4x -3y)}{x^2y^3}$$ with $x$ and $y$ positive.
    (Hint: Use AM-GM with $n=4$.)
16. Find the least value of $xy +2xz +3yz$ for positive numbers $x,y,z$, satisfying $xyz=48$.
17. Multiply $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$$ and find the minimum value of the product. Hence find the least value of $x^{-1}+y^{-1}+z^{-1}$ over the positive real numbers $x,y,z$ having a constant sum.
18. Show that among all the triangles of a given perimeter, the equilateral triangle has the largest area.
19. Show that among all the quadrilaterals of a specified perimeter, the square has the largest area.
20. Show that a quadrilateral inscribed in a circle has a larger area than any other quadrilateral with sides of the same lengths in the same order.
21. Find the length of the longest ladder that can be moved around a right-angle corner from a corridor of width $a$ to a corridor of width $b$. (Calculus by Larson, et.al.)
22. Two posts, one 12 feet high and the 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. How long a wire is needed? (Calculus by Larson, et.al.)
23. Given a line segment $AB$ and a line $\ell$ not intersecting the given line segment, find the point $P$ on $\ell$ such that the segment $AB$ subtends the greatest angle at $P$.
24. $AB$ is a diameter of a circle of radius 1. $C$ and $E$ are distinct points on the circle and on the same side of $AB$. Parallel chords $CD$ and $EF$ cut $AB$ at a $45^\circ$ angle, at points $P$ and $Q$, respectively. Prove that $PC\cdot QE+PD\cdot QF<2$. (China National 1981)
25. If the sum of the lengths of six edges of a trirectangular tetrahedron $PABC$ (i.e. $\angle APB=\angle BPC=\angle CPA= 90^\circ$) is $S$, determine its maximum volume.
    (Fifth USAMO 1976)