MathDB

1

Part of 2009 Iran MO (2nd Round)

Problems(2)

Polynomials - Iran NMO 2009 - Problem 1

Source:

9/20/2010
Let p(x) p(x) be a quadratic polynomial for which : ∣p(x)∣≤1∀x∈{−1,0,1} |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} Prove that:  ∣p(x)∣≤54∀x∈[−1,1] \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]
algebrapolynomialquadraticsalgebra proposed
Soldiers - Iran NMO 2009 - Problem 4

Source:

9/20/2010
We have a (n+2)×n (n+2)\times n rectangle and we’ve divided it into n(n+2)  1×1 n(n+2) \ \ 1\times1 squares. n(n+2) n(n+2) soldiers are standing on the intersection points (n+2 n+2 rows and n n columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a n×(n+2) n\times(n+2) rectangle (n n rows and n+2 n+2 columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left 11 and 11 to the right). Prove that n n is even.
geometryrectangleinductioncombinatorics proposedcombinatorics