Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$ Here $d(n)$ is the number of positive divisors of a positive integer

Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number \[ p - \Big\lfloor \frac{p}{q} \Big\rfloor q \] is squarefree (i.e. is not divisible by the square of a prime).

Solve in integers $20^x+13^y=2013^z$.

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$. (15 points)

We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.

Let $p{}$ be a fixed prime number. Determine the number of ordered $k$-tuples $(a_1,\ldots,a_k)$ of non-negative integers smaller than $p{}$ for which $p\mid a_1^2+\cdots+a_k^2$ where a) $k=3$ and b) $k$ is an arbitrary odd positive integer.

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.

Let be two natural numbers $ n $ and $ a. $ [b]a)[/b] Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality. $$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$ [b]b)[/b] Show that there exist only finitely such $ n\text{-tuplets} . $

For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$.

Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.

Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying: i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$; ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$ Prove that:$S=N$ I hope it hasn't posted before. :lol: :lol:

Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$. Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.

Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Is there a 30-digit number such that any number formed by its five consecutive digits is divisible by 13?

King Arthur one day had to fight the Dragon with Three Heads and Three Tails. His task became easier when he managed to find a magic sword that could, with a single blow, do one (and only one) of the following things: $\bullet$ cut off a head, $\bullet$ cut off two heads, $\bullet$ cut a tail, $\bullet$ cut off two tails. Furthermore, Fairy Morgana revealed to him the dragon's secret: $\bullet$ if a head is cut off, a new one grows, $\bullet$ if two heads are cut off nothing happens, $\bullet$ in place of a tail, two new tails are born, $\bullet$ if two tails are cut off a new head grow, $\bullet$ and the dragon dies if it loses its three heads and three tails. How many hits are needed to kill the dragon?

(a) Suppose that $x_n~(n\ge0)$ is a sequence of real numbers with the property that $x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2$ for each $n\in\mathbb N$. Prove that for each $n\in\mathbb N_0$ there exists $m\in\mathbb N_0$ such that $x_0+x_1+\ldots+x_n=\frac{m(m+1)}2$. (b) For natural numbers $n$ and $p$, we define $S_{n,p}=1^p+2^p+\ldots+n^p$. Find all natural numbers $p$ such that $S_{n,p}$ is a perfect square for each $n\in\mathbb N$.

Positive integer $\underline{a}\,\, \underline{b}\,\, \underline{c}\,\, \underline{d}\,\, \underline{r}\,\, \underline{s}\,\, \underline{t}$ has digits $a$, $b$, $c$, $d$, $r$, $s$, and $t$, in that order, and none of the digits is $0$. The two-digit numbers $\underline{a}\,\, \underline{b}$ , $\underline{b}\,\, \underline{c}$ , $\underline{c}\,\, \underline{d}$ , and $\underline{d}\,\, \underline{r}$ , and the three-digit number $\underline{r}\,\, \underline{s}\,\, \underline{t}$ are all perfect squares. Find $\underline{a}\,\, \underline{b}\,\, \underline{c}\,\, \underline{d}\,\, \underline{r}\,\, \underline{s}\,\, \underline{t}$ .

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c+d$.

Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_1,a_2,...,a_n$ are real numbers such that $a_1+a_2+\cdots +a_n$ is an integer, there exists some $i$ such that $\left|a_i-\frac{1}{2}\right|\ge f(n)$.