Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.

Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$

Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?

Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$

Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?

Suppose that $n$ is s positive integer. Determine all the possible values of the first digit after the decimal point in the decimal expression of the number $\sqrt{n^3+2n^2+n}$

Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values ​​of the number $k$ if $n=d_2d_3 + d_2d_5+d_3d_5$.

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.

[b]p1.[/b] Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle. [b]p2.[/b] Determine the greatest integer that will divide $13,511$, $13,903$ and $14,589$ and leave the same remainder. [b]p3.[/b] Suppose $A, B$ and $C$ are the angles of the triangle. Show that $\cos^2 A + \cos^2 B + \cos^2 C + 2 \cos A \cos B \cos C = 1$ [b]p4.[/b] Given the linear fractional transformation $f_1(x) =\frac{2x - 1}{x + 1}$. Define $f_{n+1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3,...$ . It can be shown that $f_{35} = f_5$. (a) Find a function $g$ such that $f_1(g(x)) = g(f_1(x)) = x$. (b) Find $f_{28}$. [b]p5.[/b] Suppose $a$ is a complex number such that $a^{10} + a^5 + 1 = 0$. Determine the value of $a^{2005} + \frac{1}{a^{2005}}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x_k=\frac{k(k+1)}{2}$ for all integers $k\ge 1$. Prove that for any integer $n \ge 10$, between the numbers $A=x_1+x_2 + \ldots + x_{n-1}$ and $B=A+x_n$ there is at least one square.

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

The set of numbers $(p, a, b, c)$ of positive integers is called [i]Sozopolian[/i] when: [b]* [/b]p is an odd prime number [b]*[/b] $a$, $b$ and $c$ are different and [b]*[/b] $ab + 1$, $bc + 1$ and $ca + 1$ are a multiple of $p$. a) Prove that each [i]Sozopolian[/i] set satisfies the inequality $p+2 \leq \frac{a+b+c}{3}$ b) Find all numbers $p$ for which there exist a [i]Sozopolian[/i] set for which the equality of the upper inequation is met.

Prove that the number $x^8+\frac{1}{x^8}$ is an integer if $x+\frac{1}{x }$ is an integer.

How many positive integers are there such that $\frac{2n^2+4n+18}{3n+3}$ is an integer?

Determine all integer solutions of the equation $x^n+y^n=1994$ where $n\geq 2$

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions: $\bullet$ All of its digits are $1$ or $2$ $\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct. For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$. Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.