2009 年

When $x^{2009}+x^{2008}+x^{2007}+\cdots+x$ is divided by $x+1$, the remainder is

If the sum of the first $n$ terms of a sequence is $n^2+2n$, then the 5th term of the sequence is

Let $a_n$ be the $n$th term of a geometric sequence. If $a_7=32$ and $a_9=8$, which of the following must be true? I. $a_1>0$ II. $a_1-a_2>0$ III. $a_2+a_3+a_4+\cdots+a_{100}>0$

Which of the following shaded regions may represent the solution of $y\le x-9$ ?

For $0^\circ\le x\le 360^\circ$, how many roots does the equation $\cos^2 x-\sin^2 x=1$ have?

Let $-90^\circ<\theta<90^\circ$. If the figure shows the graph of $y=7\sin(x^\circ+\theta)$, then

In the figure, $ABCDEFGH$ is a rectangular block. $EG$ and $FH$ intersect at $X$. $M$ is the mid-point of $BG$. If the angle between $MX$ and the plane $BCHG$ is $\theta$, then $\tan\theta=$

In the figure, $AB$ is a diameter of the circle $ABCD$. It is given that $AC$ and $BD$ intersect at $E$. If $\angle AED=\theta$, then $\dfrac{CD}{AB}=$

In the figure, $ABCD$ is a circle. If $AB=AC$, $AB\parallel DC$ and $\angle ABD=40^\circ$, then $\angle CBD=$

In the figure, $AB$ is the tangent to the circle at $A$. If $AB=20$ and $BC=50$, find the radius of the circle.