Practice (28)
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Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$?
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?
What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?
What time was it $2011$ minutes after midnight on January $1$, $2011$?
Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?
Each of the digits $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$ is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
If $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$, $r$, and $s$?
What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?
What is the product of all the roots of the equation \[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]
Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?
Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$?
Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$. What is $ab$?
What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$?
Suppose that $4^a = 5$, $5^b = 6$, $6^c = 7$, and $7^d = 8$. What is $a \cdot b\cdot c \cdot d$?
The product of the digits of positive integer $n$ is $20$, and the sum of the digits is $13$. What is the smallest possible value of $n$?
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 $j - k$?
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 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 $|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|$.
What is the area of region bounded by the graphs of $y=|x+2| -|x-2|$ and $y=|x+1|-|x-3|$?
Let $S$ be the sum of all distinct real solutions of the equation $$\sqrt{x+2015}=x^2-2015$$
Compute $\lfloor 1/S \rfloor$.
The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?
According to the standard convention for exponentiation,
$$2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536$$
If the order in which the exponentiations are performed is changed, how many other values are possible?
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]
How many ordered triples of positive integers $(x, y, z)$ satisfy $(x^y)^z = 64$?