Moldova City Olympiad 2008.

Posted on February 24, 2008 by aimingbeyond

Here are the problems for the 12th grade. I will try to post the problems for other grades soon. The problems are rather easy, for beginners mainly. However when I find time, I will try to provide their solutions.

Moldova, Chisinau, City Olympiad 2008, February 23.

Grade 12.

Problem 1.

The polynomial $P(X)=aX^3+bX^2+cX+d$ takes integer values for $X=-1,0,1,2$ . Prove that $P(X)$ is an integer for any integer $X$ .

Problem 2.

Prove that $\displaystyle 1-e^{-\frac\pi2}<\int_0^{\frac\pi2}e^{-\sin x}\textrm{d}x<\frac\pi2(1-e^{-1})$ .

Problem 3.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ , $f(x)=x^{2007}e^x\textrm{d}x$ . Prove that if $F(x)$ is a primitive for $f(x)$ then there exist $2008$ unique integers $a_0,a_1,ldots,a_{2007}$ so that $F(x)=[a_0+a_1(x-1)+a_2(x-1)^2+\ldots+a_{2007}(x-1)^{2007}]e^x+C$ .