1. 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.
  2. 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 \[ 2b^3 + 9a^2d - 7abc \leq 0. \]
  3. 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]$.
  4. 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 \[ \frac {XZ+YW}{AC + BD}. \]
  5. 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.
  6. Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $$(8x_2 - 7x_1)x_1^7 = 8,\quad x_{k + 1}x_{k - 1} - x_k^2 = \frac {x_{k - 1}^8 - x_k^8}{x_k^7x_{k - 1}^7},\, k = 2,3,\ldots.$$ 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.
  7. 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?
  8. 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$.

Post A Comment:

0 comments: