Practice (4)
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The larger of two consecutive odd integers is three times the smaller. What is their sum?
Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$. What is $ab$?
Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}$. What is the value of $a^{4} + a^{ - 4}$?
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?
Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $\frac{T}{N}$?
What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$?
A parabola with equation $y=x^2+bx+c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?
Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0$. Suppose that $a+\frac1b$ and $b+\frac1a$ are the roots of the equation $x^2-px+q=0$. What is $q$?
Let $a_1 , a_2 , ...$ be a sequence for which$a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?
The equations $2x + 7 = 3$ and $bx - 10 = - 2$ have the same solution. What is the value of $b$?
There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?
The first term of a sequence is $2005$. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the $2005^\text{th}$ term of the sequence?
The quadratic equation $x^2+mx+n=0$ has roots twice those of $x^2+px+m=0$, and none of $m,n,$ and $p$ is zero. What is the value of $\frac{n}{p}$?
Suppose that $4^a = 5$, $5^b = 6$, $6^c = 7$, and $7^d = 8$. What is $a \cdot b\cdot c \cdot d$?
Each term in the sequence that begins 13, 9, 18, $\cdots$ is the sum of three times the tens digit and two times the units digit of the previous term. What is the greatest value of any term in this sequence?
Let the sequence {$a_n$} satisfy $a_0=0$, $a_1=1$, $a_{n+2} = (n+3)a_{n+1} -(n+2)a_n$. Find whether the following equation is solvable in rational numbers:$$\sum_{i=1}^n\frac{x^i}{a_i-a_{i-1}}=-1\qquad\qquad(n \ge 2)$$
If the sum of two numbers is 4 and their difference is 2, what is their product?
The arithmetic mean of 11 numbers is 78. If 1 is subtracted from the first, 2 is subtracted from the second, 3 is subtracted from the third, and so forth, until 11 is subtracted from the eleventh, what is the arithmetic mean of the 11 resulting numbers?
What is the value of $\frac{444^2-111^2}{444-111}$ ?
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 $n$?
Real numbers a and b satisfy the equation $\frac{2a-4}{5}+\frac{3a+1}{5}=b$. What is the value of $a - b$? Express your answer as a common fraction.
If $4(a - 3) - 2(b + 5) = 14$ and $5b -a = 0$, what is the value of $a + b$?
The product of two consecutive integers is five more than their sum. What is the smallest possible sum of two such consecutive integers?