Found problems: 178
2001 Estonia Team Selection Test, 4
Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.
2020 Romania EGMO TST, P2
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.
1994 North Macedonia National Olympiad, 3
a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
2017 Saudi Arabia BMO TST, 1
Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)
1976 IMO Longlists, 41
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
2015 Romania Team Selection Tests, 3
Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .
1997 Israel National Olympiad, 3
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
2019 Dutch IMO TST, 2
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.
2015 Bulgaria National Olympiad, 1
The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.
2002 Nordic, 3
Let ${a_1, a_2, . . . , a_n,}$ and ${b_1, b_2, . . . , b_n}$ be real numbers with ${a_1, a_2, . . . , a_n}$ distinct. Show that if the product ${(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}$ takes the same value for every ${ i = 1, 2, . . . , n, }$ , then the product ${(a_1 + b_j)(a_2 + b_j) \cdot \cdot \cdot (a_n + b_j)}$ also takes the same value for every ${j = 1, 2, . . . , n, }$ .
2015 Czech-Polish-Slovak Junior Match, 6
The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.
2014 Danube Mathematical Competition, 2
Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.
2018 Bosnia and Herzegovina EGMO TST, 1
$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$
$a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$
1936 Moscow Mathematical Olympiad, 026
Find $4$ consecutive positive integers whose product is $1680$.
1992 Austrian-Polish Competition, 8
Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy:
$\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$
What values can be taken by the product
$\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?
2010 Czech And Slovak Olympiad III A, 5
On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.
1997 Abels Math Contest (Norwegian MO), 2b
Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.
1950 Moscow Mathematical Olympiad, 182
Prove that $\frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{7}{8} ... \frac{99}{100 } <\frac{1}{10}$.
2019 IMO Shortlist, A5
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
1996 Czech And Slovak Olympiad IIIA, 4
Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.
2012 Abels Math Contest (Norwegian MO) Final, 3a
Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.
2016 Lusophon Mathematical Olympiad, 1
Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?
2018 Danube Mathematical Competition, 3
Find all the positive integers $n$ with the property:
there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$
such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.
1992 Bundeswettbewerb Mathematik, 2
A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.
2004 Switzerland Team Selection Test, 7
The real numbers $a,b,c,d$ satisfy the equations:
$$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$
Prove that $abcd = 2004$.