All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.

The diagram below shows circles radius $1$ and $2$ externally tangent to each other and internally tangent to a circle radius $3$. There are relatively prime positive integers $m$ and $n$ so that a circle radius $\frac{m}{n}$ is internally tangent to the circle radius $3$ and externally tangent to the other two circles as shown. Find $m+n$. [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw(circle((8,2), 3)); draw(circle((8,1), 2)); draw(circle((8,4), 1)); draw((8,-1)--(8,5)); draw(circle((9.72,3.28), 0.86)); label("$ 2 $",(7.56,1.38),SE*labelscalefactor); label("$ 1 $",(7.6,4.39),SE*labelscalefactor); [/asy]

Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$

Positive integers $a$ and $b$ have $n$ digits each in their decimal representation. Assume that $m$ is a positive integer such that $\frac n2<m<n$ and assume that each of the leftmost $m$ digits of $a$ is equal to the corresponding digit of $b$. Prove that $$a^{\frac1n}-b^{\frac1n}<\frac1n.$$

Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.

For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.

Starting from a positive integer $n$, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of $n$, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times. There is a nice solution to this...

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[ \frac{2^n+2}{n} \] is also an integer.

Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

Let $ p_i $ denote the $ i $-th prime number. Let $ n = \lfloor\alpha^{2015}\rfloor $, where $ \alpha $ is a positive real number such that $ 2 < \alpha < 2.7 $. Prove that $$ \displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017 $$

If $ a$, $ b$, and $ c$ are positive real numbers such that $ a(b \plus{} c) \equal{} 152$, $ b(c \plus{} a) \equal{} 162$, and $ c(a \plus{} b) \equal{} 170$, then abc is $ \textbf{(A)}\ 672 \qquad \textbf{(B)}\ 688 \qquad \textbf{(C)}\ 704 \qquad \textbf{(D)}\ 720 \qquad \textbf{(E)}\ 750$

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.

Consider all nine-digit numbers, consisting of non-repeating digits from $1$ to $9$ in an arbitrary order. A pair of such numbers is called “conditional” if their sum is equal to $987654321$. (a) Prove that there exist at least two conditional pairs (noting that ($a,b$) and ($b,a$) is considered to be one pair). (b) Prove that the number of conditional pairs is odd. (G Galperin, Moscow)

[b]p1.[/b] Compute $45\times 45 - 6$. [b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers? [b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$? [b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers? [b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins? [b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo? [b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img] [b]p8.[/b] What are the last four digits of $2^{1000}$? [b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box. [b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$ [b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the number $\binom np-\left\lfloor\frac np\right\rfloor$ is divisible by $p$ for every prime number and integer $n\ge p$.

Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

Find all natural numbers $a$ such that for every positive integer $n$ the number $n(a+n)$ is not a perfect square.

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$.

Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that $|6n^2-17n-39|=p^k$

Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?

Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$