Olympiad problems

2011 Belarus Team Selection Test, 2

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]

2022 Irish Math Olympiad, 8

Tags: multiplication , number theory

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.

2010 IMO Shortlist, 1

Tags: equation , number theory , multiplication

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]

2021 Nigerian MO Round 3, Problem 4

Tags: multiplication , combinatorics

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}

2011 Brazil Team Selection Test, 2

Tags: equation , multiplication , number theory

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]

2018 EGMO, 2

Tags: number theory , multiplication

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]

1980 Putnam, B5

Tags: convexity , multiplication

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.