CGMO 2008

(a) Determine if the set $\{1,2,\ldots,96\}$ can be partitioned into 32 sets of equal size and equal sum. (b) Determine if the set $\{1,2,\ldots,99\}$ can be partitioned into 33 sets of equal size and equal sum.

Let $\varphi(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $\varphi(x)$ has three positive real roots and that $\varphi(0) < 0$ , prove that $$2b^3 + 9a^2d - 7abc \leq 0.$$

Determine the least real number $a$ greater than $1$ such that for any point $P$ in the interior of the square $ABCD$ , the area ratio between two of the triangles $PAB$ , $PBC$ , $PCD$ , $PDA$ lies in the interval $\left[\frac {1}{a},a\right]$ .

Equilateral triangles $ABQ$ , $BCR$ , $CDS$ , $DAP$ are erected outside of the convex quadrilateral $ABCD$ . Let $X$ , $Y$ , $Z$ , $W$ be the midpoints of the segments $PQ$ , $QR$ , $RS$ , $SP$ , respectively. Determine the maximum value of $$\frac {XZ+YW}{AC + BD}.$$

In convex quadrilateral $ABCD$ , $AB = BC$ and $AD = DC$ . Point $E$ lies on segment $AB$ and point $F$ lies on segment $AD$ such that $B$ , $E$ , $F$ , $D$ lie on a circle. Point $P$ is such that triangles $DPE$ and $ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $Q$ is such that triangles $BQF$ and $ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $A$ , $P$ , $Q$ are collinear.

Let $(x_1,x_2,\cdots)$ be a sequence of positive numbers such that $(8x_2 - 7x_1)x_1^7 = 8$ and $$x_{k + 1}x_{k - 1} - x_k^2 = \frac {x_{k - 1}^8 - x_k^8}{x_k^7x_{k - 1}^7} \text{ for }k = 2,3,\ldots$$

Determine real number $a$ such that if $x_1 > a$ , then the sequence is monotonically decreasing, and if $0 < x_1 < a$ , then the sequence is not monotonic.

On a given $2008 \times 2008$ chessboard, each unit square is colored in a different color. Every unit square is filled with one of the letters C, G, M, O. The resulting board is called harmonic if every $2 \times 2$ subsquare contains all four different letters. How many harmonic boards are there?

For positive integers $n$ , $f_n = \lfloor2^n\sqrt {2008}\rfloor + \lfloor2^n\sqrt {2009}\rfloor$ . Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $f_1,f_2,\ldots$ .