Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.

Find positive integers n that satisfy the following two conditions: (a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits. (b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers. (L Borisov)

Find the number of three-digit palindromes that are divisible by $3$. Recall that a palindrome is a number that reads the same forward and backward like $727$ or $905509$.

Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?

p1. The parabola $y = ax^2 + bx$, $a < 0$, has a vertex $C$ and intersects the $x$-axis at different points $A$ and $B$. The line $y = ax$ intersects the parabola at different points $A$ and $D$. If the area of triangle $ABC$ is equal to $|ab|$ times the area of ​​triangle $ABD$, find the value of $ b$ in terms of $a$ without use the absolute value sign. p2. It is known that $a$ is a prime number and $k$ is a positive integer. If $\sqrt{k^2-ak}$ is a positive integer, find the value of $k$ in terms of $a$. p3. There are five distinct points, $T_1$, $T_2$, $T_3$, $T_4$, and $T$ on a circle $\Omega$. Let $t_{ij}$ be the distance from the point $T$ to the line $T_iT_j$ or its extension. Prove that $\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}$ and $\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}$ [img]https://cdn.artofproblemsolving.com/attachments/2/8/07fff0a36a80708d6f6ec6708f609d080b44a2.png[/img] p4. Given a $7$-digit positive integer sequence $a_1, a_2, a_3, ..., a_{2017}$ with $a_1 < a_2 < a_3 < ...<a_{2017}$. Each of these terms has constituent numbers in non-increasing order. Is known that $a_1 = 1000000$ and $a_{n+1}$ is the smallest possible number that is greater than $a_n$. As For example, we get $a_2 = 1100000$ and $a_3 = 1110000$. Determine $a_{2017}$. p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume $V$. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $ b$ hours and continues again using only pump-2 for $c$ hours. If the operating cost of pump-1 is $15(a + b)$ thousand per hour and pump-2 operating cost is $4(a + c)$ thousand per hour, determine $ b$ and $c$ so that the operating costs of all pumps are minimum (express $b$ and $c$ in terms of $a$). Also determine the possible values ​​of $a$.

Solve the equation $xy =1997(x + y)$ in integers.

A subset $S$ of $ {1,2,...,n}$ is called balanced if for every $a $ from $S $ there exists some $ b $from $S$, $b\neq a$, such that $ \frac{(a+b)}{2}$ is in $S$ as well. (a) Let $k > 1 $be an integer and let $n = 2k$. Show that every subset $ S$ of ${1,2,...,n} $ with $|S| > \frac{3n}{4}$ is balanced. (b) Does there exist an $n =2k$, with $ k > 1 $ an integer, for which every subset $ S$ of ${1,2,...,n} $ with $ |S| >\frac{2n}{3} $ is balanced?

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?

Given a prime number $p > 2$. Find the least $n\in Z_+$, for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$

Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

[b]p1.[/b] Let $2^k$ be the largest power of $2$ dividing $30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1$. Find $k$. [b]p2.[/b] Let $d(n)$ be the total number of digits needed to write all the numbers from $1$ to $n$ in base $10$, for example, $d(5) = 5$ and $d(20) = 31$. Find $d(2012)$. [b]p3.[/b] Jim and TongTong play a game. Jim flips $10$ coins and TongTong flips $11$ coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins? [b]p4.[/b] There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least $150$ potatoes in the pile. What is the least number of potatoes there can be in the pile? [b]p5.[/b] Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice? [b]p6.[/b] Brett has an $ n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $ 1 \times 1 \times 1$ cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Choose two real numbers $x$ and $y$ uniformly at random from the interval $[0, 1]$. What is the probability that $x$ is closer to $1/4$ than $y$ is to $1/2$? [b]p8. [/b] In triangle $ABC$, we have $\angle BAC = 20^o$ and $AB = AC$. $D$ is a point on segment $AB$ such that $AD = BC$. What is $\angle ADC$, in degree. [b]p9.[/b] Let $a, b, c, d$ be real numbers such that $ab + c + d = 2012$, $bc + d + a = 2010$, $cd + a + b = 2013$, $da + b + c = 2009$. Find $d$. [b]p10. [/b]Let $\theta \in [0, 2\pi)$ such that $\cos \theta = 2/3$. Find $\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$. (A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values ​​of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

Let $m \in {\mathbb R}$ and $$x^2+(m-4)x+(m^2-3m+3)=0$$ equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.

Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]