Practice (4)
back to index | new
If $40q = p + \frac{p}{3}+\frac{p}{9}+\frac{p}{27}$ , what is the ratio $\frac{q}{p}$? Express your answer as a common fraction.
What is the sum of all real numbers $x$ such that $4^x - 6 \times 2^x + 8 = 0$?
Seven pounds of Mystery Meat and four pounds of Tastes Like Chicken cost \$78.00. Tastes Like Chicken costs \$3.00 more per pound than Mystery Meat. In dollars, how much does a pound of Mystery Meat cost?
Jay and Mike were walking home with heavy books in their backpacks. When Mike complained about the weight in his backpack, Jay remarked, "If I take one of your books, I will be carrying twice as many books as you will be carrying, but if you take one of my books, we'll be carrying the same number of books." How many books is Mike carrying in his backpack?
Alex added the page numbers of a book together and got a total of 888. Unfortunately, he didn't notice that one of the sheets of the book was missing with an odd page number on the front and an even page number on the back. What was the page number on the final page in the book?
The product of the integers from 1 through 7 is equal to $2^j\cdot 3^k\cdot 5 \cdot 7$ What is the value of $j - k$?
In a sequence of positive integers, every term after the first two terms is the sum of the previous two terms of the sequence. The fifth term is 2012 so what is the maximum possible value of the first term?
The figure shows the first three stages of a fractal, respectively. We must find how many circles in Stage 5 of the fractal.
Let $f(x) = x^2 + 5$, and $g(x) = 2(f(x))$. What is the greatest possible value of $f(x + 1)$ when $g(x)$ = 108?
The sum of the squares of two positive numbers is 20 and the sum of their reciprocals is 2. We must find their product.
The sum of the first $n$ terms of a sequence, $a_1 + a_2 + \cdots + a_n$, is given by the formula $S_n = n^2 + 4n + 8$ What is the value of $a_6$?
Adult tickets are \$5 and student tickets are \$2. Five times as many student tickets were sold as adult tickets for a total of \$1125. We must find the number of tickets sold.
The positive difference of the cubes of two consecutive positive integers is 111 less than 5 times the product of the two consecutive integers. We must find the sum of the two consecutive integers.
Seven consecutive positive integers have a sum of 91. So what is the largest of these integers?
We are asked to find the largest sum of calendar dates for seven consecutive Fridays in any given year.
Carol, Jane, Kim, Nancy and Vicky competed in a 400-meter race. Nancy beat Jane by 6 seconds. Carol finished 11 seconds behind Vicky. Nancy finished 2 seconds ahead of Kim, but 3 seconds behind Vicky. We are asked to find by how many seconds Kim finished ahead of Carol.
Okta stays in the sun for $16$ minutes before getting sunburned. Using a sunscreen, he can stay in the sun $20$ times as long before getting sunburned (or $320$ minutes). If he stays in the sun for $9$ minutes and then applies the sunscreen, how much longer can he remain in the sun?
If $x^2+\frac{1}{x^2}= 3$, then what is the value of $\frac{x^2}{(x^2+1)^2}$?
Mike wrote a list of 6 positive integers on his paper. The first two are chosen randomly. Each of the remaining integers is the sum of the two previous integers. We are asked to find the ratio of the fifth integer to the sum of all 6 integers.
Evaluate $\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}$.
Let $f(x) = \sqrt{2^2-x^2}$. Find the value of $f(f(f(f(f(-1)))))$.
For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?
If the numbers $x_1, x_2, x_3, x_4,$ and $x_5$ are a permutation of the numbers 1, 2, 3, 4, and 5, compute the maximum possible value of $|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|$.
Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.)
Define a sequence of positive integers $s_1, s_2, . . . , s_{10}$ to be $terrible$ if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$:
- $s_i > s_j$
- $j - i + 1$ divides the quantity $s_i + s_{i+1} + \cdots + s_j$
Determine the minimum possible value of $s_1 + s_2 + \cdots + s_{10}$ over all terrible sequences.