jump to navigation

Problem of the Day #353: Circles in Circles March 6, 2012

Posted by Alex in : potd , add a comment

Circle $O$ with radius $16$ is internally tangent to circles $A$, $B$, and $C$. Circles $A$, $B$, and $C$ do not overlap. Circle $C$ has radius $11$ and the radii of circles $A$, $B$, and $C$ (in that order) form an increasing arithmetic sequence. Find the sum of the radii of circles $A$, $B$, and $C$.

Problem of the Day #351: Points in a Circle March 4, 2012

Posted by Saketh in : potd , add a comment

Albert draws a unit circle centered at the origin, and then draws a number of points in it. Then, he adds up for each point the square of the distance to the nearest adjacent point. What is the greatest possible sum he could find?

Problem of the Day #344: Triangles from Polygon Vertices February 26, 2012

Posted by Alex in : potd , add a comment

A $24$-sided regular polygon is centered at $(0, 0)$ and at least one of its points lies on the $x$-axis. How many non-degenerate triangles formed from three vertices of the polygon contain the origin?

Problem of the Day #338: Albert’s Quadrilaterals February 20, 2012

Posted by Saketh in : potd , add a comment

Given any quadrilateral $ABCD$, Albert draws in the points $P$, $Q$, $R$, and $S$ – the midpoints of sides $BC$, $CD$, $DA$, and $AB$ respectively. He then determines the value of $$\frac{AP^2 + BQ^2 + CR^2 + DS^2}{AB^2 + BC^2 + CD^2 + DA^2}$$ What is the largest possible ratio Albert could find?

Problem of the Day #337: Ants on a Triangle February 19, 2012

Posted by Alex in : potd , add a comment

$\triangle ABC$ is a right triangle with $\angle C$ as the right angle. One ant starts at point $A$, and another ant starts at point $B$. The ants travel the vertices in the order $A \rightarrow B \rightarrow C \rightarrow A$ in an endless cycle. The midpoint of the line segment connecting the two ants forms a closed region $R$. Find the maximum possible value of $\frac{[R]}{[\triangle ABC]}$.

Problem of the Day #333: Squares within Squares February 15, 2012

Posted by Alex in : potd , add a comment

Square $ABCD$ has side length $1$. Albert chooses points $E$, $F$, $G$, and $H$ on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DA}$ respectively, such that each point divides its segment into two parts, with lengths given by the ratio $2:3$. Points $E$, $F$, $G$, and $H$ are closer to points $A$, $B$, $C$, and $D$ respectively. Find the area of square $EFGH$.

Problem of the Day #330: Incircle and Circumcircle February 12, 2012

Posted by Alex in : potd , add a comment

Find the ratio of the circumradius of a regular dodecahedron to its inradius.

Bonus: Find this ratio for any regular polygon with $n$ sides.

Problem of the Day #327: Kite in a Circle February 9, 2012

Posted by Alex in : potd , 1 comment so far

Kite $ABCD$ is inscribed in a circle of radius $5$. Line segment $AC$ passes through the center of the circle and $m\angle ACB = \frac{\pi}{6}$. Find the area of the kite.

Problem of the Day #319: Circles in Triangles in Quadrilaterals February 1, 2012

Posted by Alex in : potd , add a comment

Two triangles share a hypotenuse and do not overlap. There is a quadrilateral whose vertices are the four unique vertices of the two triangles. The inradius of one triangle is $a$ and the area of the quadrilateral is $b$ times the area of that same triangle. Find the inradius of the other triangle in terms of $a$ and $b$.

Problem of the Day #309: Reflecting a Triangle January 22, 2012

Posted by Alex in : potd , add a comment

$\triangle ABC$ has coordinates located at $(1, 1)$, $(-2, 1)$, and $(2, 0)$. When $\triangle ABC$ is reflected across the line $y=kx$, the total area covered by $\triangle ABC$ and its image is $\frac{3[ABC]}{2}$. Find $k$.