Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?

[b]p1.[/b] At $8$ AM Black Widow and Hawkeye began to move towards each other from two cities. They were planning to meet at the midpoint between two cities, but because Black Widow was driving $100$ mi/h faster than Hawkeye, they met at the point that is located $120$ miles from the midpoint. When they met Black Widow said ”If I knew that you drive so slow I would have started one hour later, and then we would have met exactly at the midpoint”. Find the distance between cities. [b]p2.[/b] Solve the inequality: $\frac{x-1}{x-2} \le \frac{x-2}{x-1}$. [b]p3.[/b] Solve the equation: $(x - y - z)^2 + (2x - 3y + 2z + 4)^2 + (x + y + z - 8)^2 = 0$. [b]p4.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is $50$ miles. Can you help Captain America to evaluate the distances between the camps. [b]p5.[/b] $N$ regions are located in the plane, every pair of them have a nonempty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p6.[/b] Numbers $1, 2, . . . , 100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1$, $a_2$, $...$ , $a_{50}$. In the second group the numbers are written in decreasing order and denoted $b_1$, $b_2$, $...$, $b_{50}$. Thus, $a_1 < a_2 < ... < a_{50}$ and $b_1 > b2_ > ... > b_{50}$. Evaluate $|a_1 - b_1| + |a_2 - b_2| + ... + |a_{50} - b_{50}|$. PS. You should use hide for answers.

Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.

Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] How many positive divisors does $72$ have? [b]E2 / L2.[/b] Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car? [b]E3 / L3.[/b] Peter wants to make fruit punch. He has orange juice ($100\%$ orange juice), tropical mix ($25\%$ orange juice, $75\%$ pineapple juice), and cherry juice ($100\%$ cherry juice). If he wants his final mix to have $50\%$ orange juice, $10\%$ cherry juice, and $40\%$ pineapple juice, in what ratios should he mix the $3$ juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime. [b]E4 / L4.[/b] Points $A, B, C$, and $D$ are chosen on a circle such that $m \angle ACD = 85^o$, $m\angle ADC = 40^o$,and $m\angle BCD = 60^o$. What is $m\angle CBD$? [b]E5.[/b] $a, b$, and $c$ are positive real numbers. If $abc = 6$ and $a + b = 2$, what is the minimum possible value of $a + b + c$? [b]E6 / L5.[/b] Circles $A$ and $B$ are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle $C$. If the distance between the centers of circles $A$ and $B$ is $20$ and the radius of circle $A$ is $16$, what is the radius of circle $B$? [b]E7.[/b] Point $P$ is inside rectangle $ABCD$. If $AP = 5$, $BP = 6$, and $CP = 7$, what is the length of $DP$? [b]E8 / L6.[/b] For how many integers $n$ is $n^2 + 4$ divisible by $n + 2$? [b]E9. [/b] How many of the perfect squares between $1$ and $10000$, inclusive, can be written as the sum of two triangular numbers? We define the $n$th triangular number to be $1 + 2 + 3 + ... + n$, where $n$ is a positive integer. [b]E10 / L7.[/b] A small sphere of radius $1$ is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a $60^o$ angle with the ground, what is the radius of the larger sphere? [b]E11 / L8.[/b] A classroom has $12$ chairs in a row and $5$ distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this? [b]E12 / L9.[/b] Let there be real numbers $a$ and $b$ such that $a/b^2 + b/a^2 = 72$ and $ab = 3$. Find the value of $a^2 + b^2$. [b]E13 / L10.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $gcd \, (x, y)+lcm \, (x, y) =x + y + 8$. [b]E14 / L11.[/b] Evaluate $\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...$ [b]E15 / L12.[/b] Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters. [b]L1.[/b] What is the sum of the positive divisors of $100$? [b]L13.[/b] Let $ABCD$ be a convex quadrilateral with $AC = 20$. Furthermore, let $M, N, P$, and $Q$ be the midpoints of $DA, AB, BC$, and $CD$, respectively. Let $X$ be the intersection of the diagonals of quadrilateral $MNPQ$. Given that $NX = 12$ and $XP = 10$, compute the area of $ABCD$. [b]L14.[/b] Evaluate $(\sqrt3 + \sqrt5)^6$ to the nearest integer. [b]L15.[/b] In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has $65\%$ of his neighbors wearing green hats, and a blue-hatted citizen has $80\%$ of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that none of the numbers $11, 111, 1111, ...$ is a square number, cube number or higher power of a natural number.

Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$ [i]Proposed by Navid Safaei[/i]

Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$

Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.

The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

Consider the addition $\begin{tabular}{cccc} & O & N & E \\ + & T & W & O \\ \hline F & O & U & R \\ \end{tabular}$ where different letters represent different nonzero digits. What is the smallest possible value of the four-digit number $FOUR$?

Find all pairs of prime numbers $p, q$ for which there exist positive integers $(m, n)$ such that $$(p+q)^m=(p-q)^n$$.

Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$, \[\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}\] Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$.

Let $a$, $b$, and $d$ be positive integers. It is known that $a+b$ is divisible by $d$ and $a\cdot b$ is divisible by $d^2$. Prove that both $a$ and $b$ are divisible by $d$.

Prove that there is no positive integers $x,y,z$ satisfying \[ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 \]

Find a sequence of natural numbers $a_i$ such that $a_i = \displaystyle\sum_{r=1}^{i+4} d_r$, where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$.

Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$?

Prove that the number $\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor$ is divisible by $10^n$ for each $n\in\mathbb N$.

There are two $3$-digit numbers which end in $99$. These two numbers are also the product of two integers which differ by $2$. What is the sum of these two numbers?

Solve the equation $2^a-5^b=3$ in positive integers $a,b$.

You are given a set of $m$ integers, all of which give distinct remainders modulo some integer $n$. Show that for any integer $k \le m$ you can split this set into $k$ nonempty groups so that the sums of elements in these groups are distinct modulo $n$. [i]Proposed by Anton Trygub[/i]

Given a real number $\alpha$ in whose decimal representation $\alpha=0,a_1a_2a_3\dots$ each decimal digit $a_i$ $(i=1,2,3,\dots)$ is a prime number. The decimal digits are arranged along the path indicated by arrows in the accompanying figure, which can be thought of as continuing infinitely to the right and downward. For each $m\geq 1$, the decimal representation of a real number $z_m$ is formed by writing before the decimal point the digit 0 and after the decimal point the sequence of digits of the $m$-th row from the top read from left to right from the adjacent arrangement. In an analogous way, for all $n\geq 1$, the real numbers $s_n$ are formed with the digits of the $n$-th column from the left to be read from top to bottom. For example, $z_3=0,a_5a_6a_7a_{12}a_{23}a_{28}\dots$ and $s_2=0,a_2a_3a_6a_{15}a_{18}a_{35}\dots$. Show: (a) If $\alpha$ is rational, then all $z_m$ and all $s_n$ are rational. (b) The converse of the statement formulated in (a) is false.

If $ x$ is a real number such that $ x^2\minus{}x$ and $ x^n\minus{}x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.

Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $