Found problems: 15460
2010 AIME Problems, 9
Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2020 SAFEST Olympiad, 2
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
2019 Polish Junior MO Finals, 1.
Let $a$, $b$ be the positive integers greater than $1$. Prove that if
$$
\frac{a}{b},\; \frac{a-1}{b-1}
$$
differ by 1, then both are integers.
2012 France Team Selection Test, 3
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:
\[a^p+b^p=p^c.\]
2023 Thailand TSTST, 4
Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.
2016 Iran MO (3rd Round), 3
Let $m$ be a positive integer. The positive integer $a$ is called a [i]golden residue[/i] modulo $m$ if $\gcd(a,m)=1$ and $x^x \equiv a \pmod m$ has a solution for $x$. Given a positive integer $n$, suppose that $a$ is a golden residue modulo $n^n$. Show that $a$ is also a golden residue modulo $n^{n^n}$.
[i]Proposed by Mahyar Sefidgaran[/i]
2014 Dutch IMO TST, 1
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
2013 Hitotsubashi University Entrance Examination, 1
Find all pairs $(p,\ q)$ of positive integers such that $3p^3-p^2q-pq^2+3q^3=2013.$
2017 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.
2017 Dutch BxMO TST, 4
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$
2011 Swedish Mathematical Competition, 1
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$
VMEO III 2006, 10.2
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
1994 Korea National Olympiad, Problem 1
Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer.
Prove that the equation cannot have five integer solutions of the form
$ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$.
Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.
2024 Lusophon Mathematical Olympiad, 6
A positive integer $n$ is called $oeirense$ if there exist two positive integers $a$ and $b$, not necessarily distinct, such that $n=a^2+b^2$.
Determine the greatest integer $k$ such that there exist infinitely many positive integers $n$ such that $n$, $n+1$, $\dots$, $n+k$ are oeirenses.
2007 Junior Balkan Team Selection Tests - Moldova, 1
The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$.
Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.
2008 Mexico National Olympiad, 2
We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices.
a) Prove that $\frac23E\le V$.
b) Can $E=V$?
2022 Czech-Polish-Slovak Junior Match, 5
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$?
2007 IMO Shortlist, 3
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$
[i]Author: Gerhard Wöginger, Netherlands[/i]
2018 Lusophon Mathematical Olympiad, 5
Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$
2023 Grand Duchy of Lithuania, 4
Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$
Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.
2020 HK IMO Preliminary Selection Contest, 13
There are $n$ different integers on the blackboard. Whenever two of these integers are chosen, either their sum or difference (possibly both) will be a positive integral power of $2$. Find the greatest possible value of $n$.
2021 Winter Stars of Mathematics, 4
Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$
[i]Flavian Georgescu[/i]
2013 Argentina Cono Sur TST, 4
Show that the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.
2017 Peru Iberoamerican Team Selection Test, P6
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that:
$$S(2^{24^{2017}}k)=S(k)$$
2020 IMO Shortlist, N6
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that
$$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$
for all $n\ge 1$
[i]Cyprus[/i]