Problems

Two concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

In base $10$ , the number $2013$ ends in the digit $3$ . In base $9$ , on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$ . For how many positive integers $b$ does the base- $b$ -representation of $2013$ end in the digit $3$ ?

The real numbers $c,b,a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$ . The quadratic $ax^2+bx+c$ has exactly one root. What is this root?

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00$ AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at $4:00$ PM. On Tuesday, when Paula wasn't there, the two helpers painted only …

In rectangle $ABCD$ , $AB=6$ , $AD=30$ , and $G$ is the midpoint of $\overline{AD}$ . Segment $AB$ is extended 2 units beyond $B$ to point $E$ , and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$ . What is the area of $BFDG$ ?

In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the …

What is the product of all the roots of the equation $$\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.$$

Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$ . The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$ ?

A circle with center $O$ has area $156\pi$ . Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$ , and point $O$ is outside $\triangle ABC$ . What is the side length of $\triangle ABC$ ?

Circle $A$ has radius $100$ . Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$ . The two circles have the same points of tangency at the beginning and end of circle $B$ 's trip. How many possible values can $r$ have?

A particular $12$ -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$ , it mistakenly displays a $9$ . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?

Rectangle $PQRS$ lies in a plane with $PQ=RS=2$ and $QR=SP=6$ . The rectangle is rotated $90^\circ$ clockwise about $R$ , then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$ ?

A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?

A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that …

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

A circle of radius $2$ is centered at $O$ . Square $OABC$ has side length $1$ . Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$ , respectively. What is the area of the shaded region in the figure, which is bounded by $BD$ , $BE$ , and the minor arc connecting $D$ and $E$ ?

Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated $45^{\circ}$ , as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original …

On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

In the sequence $2001$ , $2002$ , $2003$ , $\ldots$ , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$ . What is the $2004^\textrm{th}$ term in this sequence?

A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2$ , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$ . The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller …

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?

Suppose that $\{a_n\}$ is an arithmetic sequence with $$a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.$$ What is the value of $a_2 - a_1 ?$