AMC 10B 2009

Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

Which of the following is equal to $\dfrac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$?

Paula the painter had just enough paint for $30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for $25$ rooms. How many cans of paint did she use for the $25$ rooms?

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?

The yard is a rectangle. The two triangular flower beds sit in opposite corners, and the trapezoid fills the middle. The trapezoid's parallel sides (of length 15 and 25) are opposite sides of the rectangle, so the rectangle has length 25. The two triangles together span the width, and each has legs equal to half the difference $(25-15)/2=5$ meters.

Twenty percent less than $60$ is one-third more than what number?

Kiana has two older twin brothers. The product of their three ages is $128$. What is the sum of their three ages?

By inserting parentheses, it is possible to give the expression $2\times3 + 4\times5$ several values. How many different values can be obtained?

In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$?

Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$?

The figure shows two triangles sharing vertex $C$: triangle $ABC$ (with $A$ at lower-left, $B$ at upper-left, $C$ in the middle) and triangle $DCE$ (with $D$ at lower-right, $E$ at upper-right). We have $AB = BC = CD = CE$.

A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$?

How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?

Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

As shown below, convex pentagon $ABCDE$ has sides $AB=3$, $BC=4$, $CD=6$, $DE=3$, and $EA=7$. The pentagon is originally positioned in the plane with vertex $A$ at the origin and vertex $B$ on the positive $x$-axis. The pentagon is then rolled clockwise to the right along the $x$-axis. Which side will touch the point $x=2009$ on the $x$-axis?

On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water?

Points $A$ and $C$ lie on a circle centered at $O$, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$. What is $\dfrac{BD}{BO}$?

Five unit squares are arranged in the coordinate plane, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$?

The five unit squares form an L-shaped (staircase) region: the bottom row has squares with $x \in [0,1]$, $[1,2]$, $[2,3]$ at $y \in [0,1]$; above that at $y \in [1,2]$ are squares at $x \in [0,1]$ and $[1,2]$ (wait — let me describe from the Asymptote code). The region consists of: a $3\times1$ strip at $y\in[0,1]$ for $x\in[0,3]$, plus a $2\times1$ strip at $y\in[1,2]$ for $x\in[0,2]$ (via $(0,1)$-$(3,1)$-$(3,3)$-$(2,3)$-$(2,0)$ boundary and $(1,0)$-$(1,2)$-$(3,2)$). Actually from the Asymptote: drawing $(0,1)$-$(3,1)$-$(3,3)$-$(2,3)$-$(2,0)$ and $(1,0)$-$(1,2)$-$(3,2)$. So the staircase region is: $[0,2]\times[0,1]$, $[2,3]\times[0,3]$, $[0,2]\times... actually the region is bounded by the outer boundary. Five unit squares means total area$= 5$.

Rectangle $ABCD$ has $AB=8$ and $BC=6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $AB$ with $\overline{ME}\perp\overline{AC}$. What is the area of $\triangle AME$?

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?

Triangle $ABC$ has a right angle at $B$, $AB=1$, and $BC=2$. The angle bisector of $\angle A$ intersects side $\overline{BC}$ at $D$. What is $BD$?

What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?

A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$?

In the top view, the $2\times 2$ square has corners at $(\pm1,\pm1)$. $M = (-1,0)$ is the midpoint of the left edge. One cut goes from $(1,1)$ to $M$; another cut goes from $(1,-1)$ perpendicular to the first cut, meeting it at point $P$ (the foot of the perpendicular from $(1,-1)$ to segment $\overline{(1,1)M}$). Triangle $B$ has vertices $(1,1)$, $(1,-1)$, and $P$. Triangle $A$ contains the upper-left region and triangle $C$ the lower-left region.

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $90$ seconds, and Robert runs clockwise and completes a lap every $80$ seconds. Both start from the same line at the same time. At some random time between $10$ minutes and $11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$?

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?