2006 年

In the figure, $O$ is the origin. The equation of $AB$ is $2x+y-8=0$ and the equation of $BC$ is $2x+3y-12=0$. If $(x,y)$ is a point lying in the shaded region $OABC$ (including the boundary), then the greatest value of $x+3y+4$ is

The first negative term in the arithmetic sequence $2006,\ 1998,\ 1990,\ \dots$ is

Let $a$, $b$ and $c$ be positive integers. If $b=\sqrt{ac}$, which of the following must be true? I. $\log a^2,\ \log b^2,\ \log c^2$ is an arithmetic sequence. II. $a^3,\ b^3,\ c^3$ is a geometric sequence. III. $4^a,\ 4^b,\ 4^c$ is a geometric sequence.

For $0^\circ<x<360^\circ$, how many roots does the equation $3\cos^2x-4\cos x+1=0$ have?

If the length of a side of a regular tetrahedron is $3\text{ cm}$, then the height of the tetrahedron is

In the figure, $O$ is the centre of the circle $ABC$. If $\angle OBC=50^\circ$ and $\angle ACO=20^\circ$, then $\angle BOA=$

In the figure, $O$ is the centre of the circle. $A$ and $B$ are points lying on the circle. If $AOC$ is a straight line and $BC$ is a tangent to the circle, then the radius of the circle is

Let $O$ be the origin. If the coordinates of the points $A$ and $B$ are $(6,0)$ and $(0,6)$ respectively, then the coordinates of the in-centre of $\triangle ABO$ are

In the figure, $ABC$ is an acute-angled triangle, $AB=AC$ and $D$ is a point lying on $BC$ such that $AD$ is perpendicular to $BC$. Which of the following must be true? I. The circumcentre of $\triangle ABC$ lies on $AD$. II. The orthocentre of $\triangle ABC$ lies on $AD$. III. The centroid of $\triangle ABC$ lies on $AD$.

Consider the circle $x^2+y^2-4x+6y-40=0$. Find the slope of the diameter passing through the point $(1,2)$.