Found problems: 32
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which $n$ does it follow that the numbers on the cards are all equal?
[i]Proposed by Oleg Košik, Estonia[/i]
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$?
$\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?
$\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$
Real numbers $a,b$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac$ and $bc$?
$
\textbf{(A) }-5 \qquad
\textbf{(B) }-\frac{10}{3} \qquad
\textbf{(C) }-\frac{10}{9} \qquad
\textbf{(D) }0 \qquad
\textbf{(E) }\frac{10}{9} \qquad
$
A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?
$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$
For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that
$\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$
$\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$
$\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$
Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$
An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?