Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?

A natural number is called [i]“sozopolian”[/i], if it has exactly two prime divisors. Does there exist 12 consecutive [i]“sozopolian”[/i] numbers?

Let $k$, $a$, and $b$, be fixed integers such that $0 \le a < k$, $0 \le b < k+1$, and $a$, $b$ are not both zero. The sequence $\{T_n\}_{n \ge k}$ satisfies $T_n = T_{n-1}+T_{n-2} \pmod{n}$, $0 \le T_n < n$, $T_k = a$, and $T_{k+1} = b$. Let the decimal expression of $T_n$ form a sequence $x=\overline{0.T_kT_{k+1} \dots}$. For instance, when $k = 66, a = 5, b = 20$, we get $T_{66}=5$, $T_{67}=20$, $T_{68}=25$, $T_{69}=45$, $T_{70}=0$, $T_{71}=45, \dots$, and thus $x=0.522545045 \dots$. Prove that $x$ is irrational.

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.

Solve for integers $a,b,c$ \[ (a+b+c)^3 + \frac{1}{2} (b+c)(c+a)(a+b) = 1 - abc \]

Let $p$ be a prime and $H\subseteq \{0,1,\ldots,p-1\}$ a nonempty set. Suppose that for each element $a\in H$ there exist elements $b$, $c\in H\setminus \{a\}$ such that $b+ c-2a$ is divisible by $p$. Prove that $p<4^k$, where $k$ denotes the cardinality of $H$.

Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.

Initially, $10$ ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some $5$ numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by $1$. If in $10,000$ moves on the board a number divisible by $2023$ appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?

How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?

Let $n \ge 3$ be an integer and $A$ be a subset of the real numbers of size n. Denote by $B$ the set of real numbers that are of the form $ x \cdot y$, where $x, y \in A$ and $x\ne y$. At most how many distinct positive primes could $B$ contain (depending on $n$)?

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

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 $a^2 + b^2 + c^2$. .

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a [i]square split[/i] of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.