For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$.Is it possible to admit that the initial numbers were a) $\{2, 2, 2\}$? b) $\{3, 3, 3\}$?

A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. [i](Ahaan Rungta, 2 points)[/i]

[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$ [b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$. [b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$. [b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$. [b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question. [b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question. PS. You should use hide for answers.

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.

Let $ f(0) \equal{} f(1) \equal{} 0$ and \[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\] Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$

For an odd prime number $p$, let $S$ denote the following sum taken modulo $p$: \[ S \equiv \frac{1}{1 \cdot 2} + \frac{1}{3\cdot 4} + \ldots + \frac{1}{(p-2)\cdot(p-1)} \equiv \sum_{i=1}^{\frac{p-1}{2}} \frac{1}{(2i-1) \cdot 2i} \pmod p\] Prove that $p^2 | 2^p - 2$ if and only if $S \equiv 0 \pmod p$.

Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

$n$ consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of $n$ if the last number in the row is odd? (A Shapovalov)

Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

Find all triplets $(x,y,n)$ of positive integers which satisfy: \[ (x-y)^n=xy \]

Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.

For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$.

Find all integers $m$ and $n$ such that $\frac{m^2+n}{n^2-m}$ and $\frac{n^2+m}{m^2-n}$ are both integers.

There are positive integers $x$ and $y$ that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.

Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that \[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\] a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$. b) Prove that $\mathcal{F}$ has exactly two elements. [i]Nelu Chichirim[/i]

Let n be an integer which is greater than 1, not divisible by 1997. Let $ a_m\equal{}m\plus{}\frac{mn}{1997}$ for all m=1,2,..,1996 $ b_m\equal{}m\plus{}\frac{1997m}{n}$ for all m=1,2,..,n-1 We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995\plus{}n}$ Prove that $ c_{k\plus{}1}\minus{}c_k<2$ for all k=1,2,...,1994+n

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

Let $k$ be a positive integer and $m$ be an odd integer. Prove that there exists a positive integer $n$ such that $n^n-m$ is divisible by $2^k$.

Prove that the numbers from $1$ to $ 15$ cannot be divided into two groups: $A$ of $2$ numbers and $B$ of $13$ numbers such that the sum of the numbers in group $B$ is equal to product of numbers in group $A$.