Found problems: 49
1988 All Soviet Union Mathematical Olympiad, 471
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.
2006 IMO Shortlist, 7
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
2013 IMAC Arhimede, 2
For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.
1954 Putnam, B7
Let $a>0$. Show that
$$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$
lies between $e^a$ and $e^{a+1}.$
2007 Germany Team Selection Test, 3
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
1957 Putnam, A2
Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.
1976 IMO Longlists, 47
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
2006 Greece JBMO TST, 2
Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.
2013 Danube Mathematical Competition, 3
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
2018 Junior Regional Olympiad - FBH, 5
Find all integers $x$ and $y$ such that $2^x+1=y^2$
2017 Balkan MO Shortlist, N3
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.
2017 India PRMO, 5
Let $u, v,w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.
1984 All Soviet Union Mathematical Olympiad, 379
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
2007 Germany Team Selection Test, 3
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
1980 Putnam, B1
For which real numbers $c$ is
$$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$
for all real $x?$
2015 Dutch IMO TST, 5
For a positive integer $n$, we de ne $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$.
1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer.
2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.
2016 Dutch IMO TST, 2
Determine all pairs $(a, b)$ of integers having the following property:
there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.
2005 IMO Shortlist, 4
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
1982 All Soviet Union Mathematical Olympiad, 329
a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$.
b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?
2009 IMO Shortlist, 7
Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square.
[i]Proposed by Mongolia[/i]
2015 Indonesia MO Shortlist, N4
Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$.
(a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$.
(b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?
1958 February Putnam, A3
Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.
2009 Federal Competition For Advanced Students, P2, 1
If $x,y,K,m \in N$, let us define:
$a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$,
Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$
1982 All Soviet Union Mathematical Olympiad, 341
Prove that the following inequality is valid for the positive $x$:
$$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$
1990 Mexico National Olympiad, 3
Show that $n^{n-1}-1$ is divisible by$ (n-1)^2$ for $n > 2$.