Found problems: 15460
2011 AIME Problems, 12
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 Romanian Master of Mathematics, 5
Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that
$$\displaystyle P(x)=R(T(x))$$
2008 Peru IMO TST, 6
We say that a positive integer is happy if can expressed in the form $ (a^{2}b)/(a \minus{} b)$ where $ a > b > 0$ are integers. We also say that a positive integer $ m$ is evil if it doesn't a happy integer $ n$ such that $ d(n) \equal{} m$. Prove that all integers happy and evil are a power of $ 4$.
2025 Junior Macedonian Mathematical Olympiad, 3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
2018 Singapore MO Open, 5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
1931 Eotvos Mathematical Competition, 2
Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.
1984 IMO Longlists, 56
Let $a, b, c$ be nonnegative integers such that $a \le b \le c, 2b \neq a + c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d, e$, and $f$ such that $d \le e \le f, f \neq c$, and such that $a^2+b^2+c^2 = d^2 + e^2 + f^2$?
2021 Germany Team Selection Test, 2
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
1999 Brazil Team Selection Test, Problem 1
Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.
2013 Peru IMO TST, 5
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
2016 Korea Junior Math Olympiad, 5
$n \in \mathbb {N^+}$
Prove that the following equation can be expressed as a polynomial about $n$.
$$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$
2024 VJIMC, 4
Let $p>2$ be a prime and let
\[\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}.\]
Prove that
\[\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.\]
2012 Chile National Olympiad, 2
Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$
2018 Rioplatense Mathematical Olympiad, Level 3, 5
Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$
is integer for every integer $d\ge 0$
2021 Purple Comet Problems, 13
Find the greatest prime number $p$ such that $p^3$ divides $$\frac{122!}{121}+ 123!:$$
2023 Durer Math Competition Finals, 6
In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
1995 Vietnam Team Selection Test, 3
Find all integers $ a$, $ b$, $ n$ greater than $ 1$ which satisfy
\[ \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995}
\]
2010 Belarus Team Selection Test, 4.3
a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$
b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich)
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2018 Iran Team Selection Test, 5
Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power:
$$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$
[i]Proposed by Navid Safaei[/i]
2007 Thailand Mathematical Olympiad, 4
Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.
2013 AIME Problems, 13
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
2017 International Olympic Revenge, 1
Let $f(x)$ be the distance from $x$ to the nearest perfect square. For example, $f(\pi) = 4 - \pi$. Let $\alpha = \frac{3 + \sqrt{5}}{2}$ and let $m$ be an integer such that the sequence $a_n = f(m \; \alpha^n)$ is bounded. Prove that either $m=k^2$ or $m = 5k^2$ for some integer $k$.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil[/i].
2008 Bulgarian Autumn Math Competition, Problem 9.3
Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.
2000 Irish Math Olympiad, 3
For each positive integer $ n$ find all positive integers $ m$ for which there exist positive integers $ x_1<x_2<...<x_n$ with:
$ \frac{1}{x_1}\plus{}\frac{2}{x_2}\plus{}...\plus{}\frac{n}{x_n}\equal{}m.$