Problems

There are a lot of problems here. Just do the ones that interest you.

1. Prove that for any $a,b,c>0$,
   $$(a+b)(b+c)(c+a) \ge 8abc,$$
   and determine when equality holds.

2. Prove $n! < \left( \frac{n+1}{2} \right)^n$ for all integers $n > 1$.

3. Prove that the sum of the legs of a right triangle never exceeds $\sqrt{2}$ times the hypotenuse of the triangle.

4. Prove $2 \sqrt{x} \ge 3-1/x$ for $x>0$.

5. Prove that if $a>b>0$ and $n \ge 1$ is an integer, then
   $$a^n-b^n > n(a-b)(ab)^{(n-1)/2}.$$

6. Let $E$ be the ellipsoid
   $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$
   for some $a,b,c>0$. Find, in terms of $a$, $b$, and $c$, the volume of the largest rectangular box that can fit inside $E$, with faces parallel to the coordinate planes.

7. Among all planes passing through a fixed point $(a,b,c)$ with $a,b,c>0$ and meeting the positive parts of the three coordinate axes, find the one such that the tetrahedron bounded by it and the coordinate planes has minimal area.

8. Among all rectangular boxes with volume $V$, find the one with smallest surface area.

9. Now consider ``open boxes,'' with only five faces. Again find the one with smallest surface area with a given volume $V$.

10. Let $T$ be the tetrahedron with vertices $(0,0,0)$, $(a,0,0)$, $(0,b,0)$, and $(0,0,c)$ for some $a,b,c>0$. Let $V$ be the volume of $T$, and let $\ell$ be the sum of the lengths of the six edges of $T$. Prove that
    $$V \le \frac{\ell^3}{6(3+3\sqrt{2})^3}.$$

11. Let $g=\sqrt[n]{a_1\dots a_n}$ be the geometric mean of the numbers $a_1,\dots,a_n>0$. Prove that
    $$(1+a_1)(1+a_2)\dots(1+a_n) \ge (1+g)^n.$$

12. Suppose $x,y,z>0$ and $x+y+z=1$. Prove that
    $$\left(1 +\frac{1}{x} \right) \left(1 +\frac{1}{y} \right) \left(1 +\frac{1}{z} \right) \ge 64.$$

13. Show that one can derive the AM-GM inequality for positive numbers from Jensen's inequality with $f(x)=-\log x$.

14. Prove $x^x \ge \left(\frac{x+1}{2}\right)^{x+1}$ for $x>0$. (Hint: the function $x \log x$ is convex on $(0,\infty)$.)

15. Use Jensen's inequality to show that among all convex $n$-gons inscribed in a fixed circle, the regular $n$-gons have the largest perimeter.

16. Given $a,b,c,p,q,r>0$ with $p+q+r=1$, prove
    $$a+b+c \ge a^p b^q c^r + a^r b^p c^q + a^q b^r c^p.$$

17. Show that by taking some of the $a_i$ to be equal in the AM-GM inequality, one can deduce the weighted AM-GM inequality at least in the case where the weights are nonnegative rational numbers. (To deduce from this the general weighted AM-GM inequality, one can then use a limit argument.) Can one similarly deduce the weighted power mean inequality and weighted Jensen's inequality from the unweighted versions?

18. Prove that if $a,b,c$ are sides of a triangle, then
    $$(a+b-c)^a (b+c-a)^b (c+a-b)^c \le a^a b^b c^c.$$

19. Given $a,b,c,d>0$ such that $(a^2+b^2)^3=c^2+d^2$, prove
    $$\frac{a^3}{c} + \frac{b^3}{d} \ge 1.$$

20. What well-known inequality does one obtain by taking only the end terms in Maclaurin's inequality?

21. Prove
    $$(bc+ca+ab)(a+b+c)^4 \le 27(a^3+b^3+c^3)^2$$
    for $a,b,c \ge 0$.

22. Prove that if $x,y,z,a,b,c>0$, then
    $$\frac{x^4}{a^3} + \frac{y^4}{b^3} + \frac{z^4}{c^3} \ge \frac{(x+y+z)^4}{(a+b+c)^3}.$$

23. Prove that if $a,b,c$ are sides of a triangle, then
    $$a^2 b(a-b) +b^2 c(b-c) + c^2 a(c-a) \ge 0.$$

24. Derive Chebychev's inequality from the rearrangement inequality.

25. Derive the 3-sequence Chebychev inequality from the 2-sequence Chebychev inequality.

26. Suppose that $0 \le \theta_1,\dots,\theta_n \le \pi/2$ and $\theta_1+\dots+\theta_n=2\pi$. Prove that
    $$4 \le \sin(\theta_1)+\dots+\sin(\theta_n) \le n \sin(2\pi/n).$$

27. Show that Jensen's inequality is a special case of the Hardy-Littlewood-Polyà majorization inequality.

28. What is the geometric meaning of Minkowski's inequality when $r=2$ and $n=3$?

29. (IMO 1975/1) Let $x_i,y_i$ ( $i=1,2,\dots,n$) be real numbers such that
    $$x_1 \ge x_2 \ge \dots \ge x_n$$   and$$\qquad y_1 \ge y_2 \ge \dots \ge y_n.$$
    Prove that if $z_1,z_2,\dots,z_n$ is any permutation of $y_1,y_2,\dots,y_n$, then
    $$\sum_{i=1}^n (x_i-y_i)^2 \le \sum_{i=1}^n (x_i-z_i)^2.$$

30. (USAMO 1977/5) Suppose $0<p<q$, and $a,b,c,d,e \in [p,q]$. Prove that
    $$(a+b+c+d+e) \left(\frac1a +\frac1b + \frac1c +\frac1d +\frac 1e\right) \le 25 + 6 \left(\sqrt{\frac{p}{q}} +\sqrt{\frac{q}{p}} \right)^2$$
    and determine when there is equality.

31. (IMO 1964/2) Prove that if $a,b,c$ are sides of a triangle, then
    $$a^2 (b+c-a) + b^2 (c+a-b) + c^2(a+b-c) \le 3abc.$$

32. (USAMO 1981/5) If $x$ is a positive real number, and $n$ is a positive integer, prove that
    $$\lfloor nx \rfloor \ge \frac{\lfloor x \rfloor}{1} + \frac{\lfloor 2x \rfloor}{2} + \dots + \frac{\lfloor nx \rfloor}{n},$$
    where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$.

33. Make your own inequality problems and give them to your friends (or enemies, depending on the difficulty!)

Many of the problems above were drawn from notes from the U.S. training session for the International Mathematics Olympiad. Others are from the USSR Olympiad Problem Book. Many of the inequalities themselves are treated in the book ``Inequalities'' by Hardy, Littlewood, and Polyà, which is a good book for further reading.

©Berkeley Math Circle