IrMO 1992

How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$ $$x^4+y^4+z^4=33,$$ $$xyz=-4?$$

Let $A$ be a nonempty set with $n$ elements. Find the number of ways of choosing a pair of subsets $(B,C)$ of $A$ such that $B$ is a nonempty subset of $C$ .

In a triangle $ABC$ , the points $A'$ , $B'$ and $C'$ on the sides opposite $A$ , $B$ and $C$ , respectively, are such that the lines $AA'$ , $BB'$ and $CC'$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB'|\cdot |BC'|\cdot |CA'|$ divided by the area of the triangle $A'B'C'$ .

Let $ABC$ be a triangle such that the coordinates of the points $A$ and $B$ are rational numbers. Prove that the coordinates of $C$ are rational if, and only if, $\tan A$ , $\tan B$ , and $\tan C$ , when defined, are all rational numbers.

If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$ , etc.. If $a_k$ consists of a single digit, for some $k\ge 1$ , $a_k$ is called a digital root of $a_1$ . It is easy to check that every positive integer has a unique root. $($ For example, if $a_1=24378$ , then $a_2=1344$ , $a_3=48$ , $a_4=32$ , $a_5=6$ , and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$ .

A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.

If, for $k=1,2,\dots ,n$ , $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$