IrMO 2010

Find the least $k$ for which the number $2010$ can be expressed as the sum of the squares of $k$ integers.

Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$ . Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $$|AD|=|DM|=2|DN|,$$ where $D$ is the intersection point of the lines $MN$ and $AP$ . Show that $|AC|=|BC|$ .

Suppose $x,y,z$ are positive numbers such that $x+y+z=1$ . Prove that (a) $xy+yz+xz\ge 9xyz$ ; (b) $xy+yz+xz<\frac{1}{4}+3xyz$ ;

The country of Harpland has three types of coins: green, white and orange. The unit of currency in Harpland is the shilling. Any coin is worth a positive integer number of shillings, but coins of the same color may be worth different amounts. A set of coins is stacked in the form of an equilateral triangle of side $n$ coins, as shown below for the case of $n=6$ .

The stacking has the following properties:

(a) no coin touches another coin of the same color;

(b) the total worth, in shillings, of the coins lying on any line parallel to one of the sides of the triangle is divisible by by three.

Prove that the total worth in shillings of the green coins in the triangle is divisible by three.

Find all polynomials $f(x)=x^3+bx^2+cx+d$ , where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$ .

For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$

In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$ . Find the length of $AD$ .

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$ .

Suppose $a,b,c$ are the side lengths of a triangle $ABC$ . Show that $$x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)}$$ are the side lengths of an acute-angled triangle $XYZ$ , with the same area as $ABC$ , but with a smaller perimeter, unless $ABC$ is equilateral.