Found problems: 49
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]
Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies
\[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\]
Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?
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 $n$.
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]
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 $n$.
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]
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 $D_n = 3^k$.
Let $a, b$ and $c$ be integers such that $a + b + c = 0$.
(a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$.
(b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$.
.
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 $e$.
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 $A_{km} (x)=B_k(y)$
Find all integers $x$ and $y$ such that $2^x+1=y^2$
a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even.
b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.)
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 $m$?
Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.
A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation
$$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$
And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$
.
For which real numbers $c$ is
$$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$
for all real $x?$
Show that $n^{n-1}-1$ is divisible by$ (n-1)^2$ for $n > 2$.
a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$).
b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.