Found problems: 15460
2012 India PRMO, 4
The letters $R, M$, and $O$ represent whole numbers. If $R \times M \times O = 240$, $R \times O + M =46$ and $R + M \times O = 64$, what is the value of $R + M + O$?
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Let's call the "Soros product" of two different numbers, $a$ and $b$, the number $a + b + ab$. Is it possible, based on numbers $1$ and $4$, after repeated application of this operation to the already obtained products, to obtain:
a) the number $1999$?
b) the number $2000$?
2001 Turkey Team Selection Test, 3
For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$
$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
2014 CHMMC (Fall), 3
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win?
2001 China Team Selection Test, 2
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
2010 Contests, 4
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.
(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.
Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.
2023 India National Olympiad, 3
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]
[i]Proposed by Sutanay Bhattacharya[/i]
2015 Romania Team Selection Tests, 3
If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that :
[b](a)[/b] $f(n)<3\sqrt{n}$ for all positive integers $n$ .
[b](b)[/b] If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.
2015 South East Mathematical Olympiad, 8
For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?
2008 Grigore Moisil Intercounty, 4
Let $ n$ be a positive integer, and $ k\leq n\minus{}1$, $ k\in \mathbb{N}$. Denote $ a_k\equal{}k!(1\plus{}\frac12\plus{}\frac13\plus{}\cdots\plus{}\frac1k)$.
Prove that the number $ k! \cdot\left[\binom{n\minus{}1}{k}\minus{}(\minus{}1)^k\right]\plus{}(\minus{}1)^k\cdot a_k \cdot n$ is divisible by $ n^2$.
1974 Miklós Schweitzer, 4
Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic.
[i]L. Lovasz, J. Pelikan[/i]
1995 Irish Math Olympiad, 5
For each integer $ n$ of the form $ n\equal{}p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1\equal{}d_1<d_2<...<d_{15}<d_{16}\equal{}n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9\minus{}d_8 \not\equal{} 22$.
2022 BMT, 5
Compute the last digit of $(5^{20}+2)^3.$
2010 Indonesia TST, 1
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $
2023 Philippine MO, 7
A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.
2010 Contests, 4
Find all positive integers $n$ which satisfy the following tow conditions:
(a) $n$ has at least four different positive divisors;
(b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]
2017 Polish Junior Math Olympiad Second Round, 4.
Do numbers $x_1, x_2, \ldots, x_{99}$ exist, where each of them is equal to $\sqrt{2}+1$ or $\sqrt{2}-1$, and satisfy the equation \[x_1x_2+x_2x_3+x_3x_4+\ldots+x_{98}x_{99}+x_{99}x_1=199\,?\] Justify your answer.
2017 ISI Entrance Examination, 7
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:
(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and
(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
2025 Belarusian National Olympiad, 9.3
Let $a_1,a_2,a_3,\ldots$ be a sequence of all composite positive integers in increasing order. A sequence $b_1,b_2,b_3,\ldots$ is given for all positive integers $i$ by equation $$b_i=ia_1^2+(i-1)a_2^2+\ldots+2a_{i-1}^2+a_i^2$$
What is the maximum amount of consecutive elements of sequence $b_1,b_2,b_3,\ldots$ which can be divisible by $3$?
[i]M. Shutro[/i]
2016 CentroAmerican, 5
We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.
2024 Kyiv City MO Round 1, Problem 3
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win?
[i]Proposed by Mykhailo Shtandenko[/i]
2021 China Team Selection Test, 4
Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$
Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.
2013 Greece Team Selection Test, 1
Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$
1994 Iran MO (2nd round), 3
Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that
\[m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).\]
Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$
2012 Finnish National High School Mathematics Competition, 3
Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$