2013 AMC 10B

What is the value of the following expression? $$
\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}
$$

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green’s steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

On a particular January day, the high temperature in Lincoln, Nebraska, was $16$ degrees higher than the low temperature, and the average of the high and low temperatures was $3$ degrees. What was the low temperature in Lincoln that day (in degrees)?

When counting from $3$ to $201,$ the number $53$ is in position $51.$ When counting backward from $201$ to $3,$ the number $53$ is in position $n.$ What is $n?$

Positive integers $a$ and $b$ are each less than $6.$ What is the smallest possible value of the following expression? $$
2 \cdot a - a \cdot b
$$

The average age of $33$ fifth-graders is $11.$ The average age of $55$ of their parents is $33.$ What is the average age of all of these parents and fifth-graders?

Six points are equally spaced around a circle of radius $1.$ Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?

Ray’s car averages $40$ miles per gallon of gasoline, and Tom’s car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars’ combined rate of miles per gallon of gasoline?

Three positive integers are each greater than $1,$ have a product of $ 27000 ,$ and are pairwise relatively prime. What is their sum?

A basketball team’s players were successful on $50\%$ of their two-point shots and $40\%$ of their three-point shots, which resulted in $54$ points. They attempted $50\%$ more two-point shots than three-point shots. How many three-point shots did they attempt?

Real numbers $x$ and $y$ satisfy the equation $$
x^2+y^2=10x-6y-34.
$$ What is $x+y?$

Let $ S $ be the set of sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying “$1$”, so Blair follows by saying “$1,$ $2$”. Jo then says “$1,$ $2,$ $3$”, and so on. What number is spoken in position $53$?

Define $ a\clubsuit b=a^2b-ab^2 .$ Which of the following describes the set of points $ (x, y) $ for which $ x\clubsuit y=y\clubsuit x ?$

A wire is cut into two pieces, one of length $a$ and the other of length $b.$ The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}?$

In triangle $\triangle ABC,$ medians $AD$ and $CE$ intersect at $P,$ $PE=1.5,$ $PD=2,$ and $DE=2.5.$ What is the area of $AEDC?$

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3.$ How many integers less than $2013$ but greater than $1000$ have this property?

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

The number $2013$ is expressed in the form $$
2013 = \frac {a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},
$$ where $$
a_1 \ge a_2 \ge \cdots \ge a_m
$$ and $$
b_1 \ge b_2 \ge \cdots \ge b_n
$$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|?$