Found problems: 7
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which
$$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$
Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.
In the multiplication magic square below, $l, m, n, p, q, r, s, t, u$ are positive integers. The product of any three numbers in any row, column or diagonal is equal to a constant $k$, where $k$ is a number between $11, 000$ and $12, 500$. Find the value of $k$.
\begin{tabular}{|l|l|l|}
\hline
$l$ & $m$ & $n$ \\
\hline
$p$ & $q$ & $r$ \\
\hline
$s$ & $t$ & $u$ \\
\hline
\end{tabular}
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
Consider the set
\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]
[list=a]
[*]Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different.
[*]For every integer $x \geq 2$ let $f(x)$ denote the minimum integer such that $x$ can be written as the
product of $f(x)$ elements of $A$, which are not necessarily different.
Prove that there exist infinitely many pairs $(x,y)$ of integers with $x\geq 2$, $y \geq 2$, and \[f(xy)<f(x)+f(y).\] (Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$).
[/list]
8. The Equation [i]AB[/i] X [i]CD[/i] = [i]EFGH[/i], where each of the letters [i]A[/i], [i]B[/i], [i]C[/i], [i]D[/i], [i]E[/i], [i]F[/i], [i]G[/i], [i]H[/i] represents a different digit and the values of [i]A[/i], [i]C[/i] and [i]E[/i] are all nonzero, has many solutions, e.g., 46 X 85 =3910. Find the smallest value of the four-digit number [i]EFGH[/i] for which there is a solution.