Found problems: 49
(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ?
(ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?
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.
Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and
$$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$
(Turkey)
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
Prove that the following inequality is valid for the positive $x$:
$$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$
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.
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 $D_n = 3^k$.
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 $(c, d) = 1$?
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]
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]
Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$.
D. Bazylev
Solve the following equation for real values of $x$:
\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.
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 $e^{a+1}.$
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
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 $n$.
In the set of all positive real numbers define the operation $a * b = a^b$ .
Find all positive rational numbers for which $a * b = b * a$.
Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$.
(D. Bazylev)
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.
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]
Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy?
Proposed by [i]Borche Joshevski, Macedonia[/i]
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]
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.