MathDB

1

Part of 2008 Saint Petersburg Mathematical Olympiad

Problems(3)

St Petersburgh 2008 #1

Source:

7/23/2011
The graph y=x2+ax+by=x^2+ax+b intersects any of the two axes at points AA, BB, and CC. The incenter of triangle ABCABC lies on the line y=xy=x. Prove that a+b+1=0a+b+1=0.
geometryincenterconicsparabolacircumcirclegeometry proposed
Negative value

Source: St Petersburg 2008 10th grade #1

7/28/2011
Replacing any of the coefficients of quadratic trinomial f(x)=ax2+bx+cf(x)=ax^2+bx+c with an 11 will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
quadraticsgeometrygeometric transformationfunctionalgebra proposedalgebra
Coloring square

Source: St Petersburg Olympiad 2008, Grade 11, P1

8/30/2017
We color some cells in 10000×1000010000 \times 10000 square, such that every 10×1010 \times 10 square and every 1×1001 \times 100 line have at least one coloring cell. What minimum number of cells we should color ?
combinatorics