Found problems: 178
2020 Brazil Team Selection Test, 3
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
2021 Polish Junior MO First Round, 5
Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.
2018 JBMO Shortlist, A4
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find:
a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$
b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.
1987 Polish MO Finals, 5
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).
2008 Postal Coaching, 3
Prove that there exists an in nite sequence $<a_n>$ of positive integers such that for each $k \ge 1$
$(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.
2007 Singapore Junior Math Olympiad, 4
The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.
2022 Indonesia TST, N
Let $n$ be a natural number, with the prime factorisation
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
\[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.
1991 Bundeswettbewerb Mathematik, 1
Given $1991$ distinct positive real numbers, the product of any ten of these numbers is always greater than $1$. Prove that the product of all $1991$ numbers is also greater than $1$.
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a finite set of positive real numbers satisfying the property:
[i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i]
Show that the product of all elements in $A$ is equal to $1$.
1964 Dutch Mathematical Olympiad, 5
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$:
(a) $g_{n+1}\le g_n$
(b) $g_{10}= 000$.
1987 IMO Longlists, 34
(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$
(b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$
[i]Proposed by Hungary.[/i]
1970 IMO Shortlist, 4
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
1936 Moscow Mathematical Olympiad, 026
Find $4$ consecutive positive integers whose product is $1680$.
1965 Dutch Mathematical Olympiad, 1
We consider the sequence $t_1,t_2,t_3,...$ By $P_n$ we mean the product of the first $n$ terms of the sequence. Given that $t_{n+1} = t_n \cdot t_{n+2}$ for each $n$, and that $P_{40} = P_{80} = 8$. Calculate $t_1$ and $t_2$.
2020 Thailand TST, 3
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
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 $\frac{1}{9}$
2020 Cono Sur Olympiad, 5
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have.
2015 Caucasus Mathematical Olympiad, 1
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
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 IMO Longlists, 47
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]
2019 Jozsef Wildt International Math Competition, W. 55
Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$
2014 Regional Olympiad of Mexico Center Zone, 1
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.
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 $2\sqrt{n}$.
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 ${j = 1, 2, . . . , n, }$ .
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.