Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]

At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of $2019$ other selected numbers and a) the selected numbers are not necessarily different? b) the selected numbers are pairwise different?

Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $

If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$ [u]Babis[/u] [b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]

Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

$p > 3$ is a prime. Find all integers $a$, $b$, such that $a^2 + 3ab + 2p(a+b) + p^2 = 0$.

Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

Let $a_n$ be the number of unordered sets of three distinct bijections $f, g, h : \{1, 2, ..., n\} \to \{1, 2, ..., n\}$ such that the composition of any two of the bijections equals the third. What is the largest value in the sequence $a_1, a_2, ...$ which is less than $2021$?

You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $ 1$ m/s. Thereafter, when exactly $ 1$ second has passed from when Al last changed its speed, Al’s speed instantaneously becomes $1$ m/s faster, so that Al runs at a constant speed of $k$ m/s in the kth second of the race. (Start counting seconds at $ 1$). Suppose Al beats you by exactly $1$ second. How many meters was the race?

Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

Prove that the sum of the $ \text{2005-th} $ powers of three pairwise distinct complex numbers is the imaginary unit if their modulus are equal and the sum of these numbers is the imaginary unit.

[b]p1.[/b] Ten math students take a test, and the average score on the test is $28$. If five students had an average of $15$, what was the average of the other five students' scores? [b]p2.[/b] If $a\otimes b = a^2 + b^2 + 2ab$, find $(-5\otimes 7) \otimes 4$. [b]p3.[/b] Below is a $3 \times 4$ grid. Fill each square with either $1$, $2$ or $3$. No two squares that share an edge can have the same number. After filling the grid, what is the $4$-digit number formed by the bottom row? [img]https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png[/img] [b]p4.[/b] What is the angle in degrees between the hour hand and the minute hand when the time is $6:30$? [b]p5.[/b] In a small town, there are some cars, tricycles, and spaceships. (Cars have $4$ wheels, tricycles have $3$ wheels, and spaceships have $6$ wheels.) Among the vehicles, there are $24$ total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town? [b]p6.[/b] You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails? [b]p7.[/b] In the below diagram, $\angle ABC$ and $\angle BCD$ are right angles. If $\overline{AB} = 9$, $\overline{BD} = 13$, and $\overline{CD} = 5$, calculate $\overline{AC}$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png[/img] [b]p8.[/b] Out of $100$ customers at a market, $80$ purchased oranges, $60$ purchased apples, and $70$ purchased bananas. What is the least possible number of customers who bought all three items? [b]p9.[/b] Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following: Francis: Fred ate the cookies. Fred: Ted did not eat the cookies. Ted: Francis is lying. If exactly one of them is telling the truth, who ate all the cookies? [b]p11.[/b] Let $ABC$ be a triangle with a right angle at $A$. Suppose $\overline{AB} = 6$ and $\overline{AC} = 8$. If $AD$ is the perpendicular from $A$ to $BC$, what is the length of $AD$? [b]p12.[/b] How many three digit even numbers are there with an even number of even digits? [b]p13.[/b] Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line. [b]p14.[/b] A man $5$ foot, $10$ inches tall casts a $14$ foot shadow. $20$ feet behind the man, a flagpole casts ashadow that has a $9$ foot overlap with the man's shadow. How tall (in inches) is the flagpole? [b]p15.[/b] Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has $n$ balls and 3 divides $n$, then he throws away a third of the balls. If $3$ does not divide $n$ but $2$ divides $n$, then he throws away half of them. If neither $3$ nor $2$ divides $n$, he stops throwing away the balls. If he began with $1458$ balls, after how many steps does he stop throwing away balls? [b]p16.[/b] Oski has $50$ coins that total to a value of $82$ cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter). [b]p17.[/b] Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC$. Suppose $\overline{MA} = \overline{MB} = \overline{MC} = 2$ and $\angle ACB = 30^o$. Find the area of the triangle. [b]p18.[/b] A spirited integer is a positive number representable in the form $20^n + 13k$ for some positive integer $n$ and any integer $k$. Determine how many spirited integers are less than $2013$. [b]p19. [/b]Circles of radii $20$ and $13$ are externally tangent at $T$. The common external tangent touches the circles at $A$, and $B$, respectively where $A \ne B$. The common internal tangent of the circles at $T$ intersects segment $AB$ at $X$. Find the length of $AX$. [b]p20.[/b] A finite set of distinct, nonnegative integers $\{a_1, ... , a_k\}$ is called admissible if the integer function $f(n) = (n + a_1) ... (n + a_k)$ has no common divisor over all terms; that is, $gcd \left(f(1), f(2),... f(n)\right) = 1$ for any integer$ n$. How many admissible sets only have members of value less than $10$? $\{4\}$ and $\{0, 2, 6\}$ are such sets, but $\{4, 9\}$ and $\{1, 3, 5\}$ are not. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.

Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?

Let $k$ be a positive integer and $a_1, a_2,... , a_k$ be nonnegative real numbers. Initially, there is a sequence of $n \geq k$ zeros written on a blackboard. At each step, Nicole chooses $k$ consecutive numbers written on the blackboard and increases the first number by $a_1$, the second one by $a_2$, and so on, until she increases the $k$-th one by $a_k$. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among $a_1, a_2, . . . , a_k$ are equal.

Prove that the polynomial $x^n+4$ can be expressed as a product of two polynomials (each with degree less than $n$) with integer coefficients, if and only if $n$ is divisible by $4$.

If $a,b,c,d$ are positive reals such that $a^2+b^2=c^2+d^2$ and $a^2+d^2-ad=b^2+c^2+bc$, find the value of $\frac{ab+cd}{ad+bc}$

[u]Set 1[/u] [b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$. [b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? [b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$? [u]Set 2[/u] [b]B4.[/b] Every day at Andover is either sunny or rainy. If today is sunny, there is a $60\%$ chance that tomorrow is sunny and a $40\%$ chance that tomorrow is rainy. On the other hand, if today is rainy, there is a $60\%$ chance that tomorrow is rainy and a $40\%$ chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is $n$? [b]B5.[/b] In the diagram below, what is the value of $\angle DD'Y$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png[/img] [b]B6.[/b] Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. [u]Set 3[/u] [b]B7.[/b] Let $P$ be a point on side $AB$ of square $ABCD$ with side length $8$ such that $PA = 3$. Let $Q$ be a point on side $AD$ such that $P Q \perp P C$. The area of quadrilateral $PQDB$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]B8.[/b] Jessica and Jeffrey each pick a number uniformly at random from the set $\{1, 2, 3, 4, 5\}$ (they could pick the same number). If Jessica’s number is $x$ and Jeffrey’s number is $y$, the probability that $x^y$ has a units digit of $1$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]B9.[/b] For two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane, we define the taxicab distance between them as $|x_1 - x_2| + |y_1 - y_2|$. For example, the taxicab distance between $(-1, 2)$ and $(3,\sqrt2)$ is $6-\sqrt2$. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? [u]Set 4[/u] [b]B10.[/b] Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the numbers that he multiplied together to get $276$? [b]B11.[/b] Let $ABCD$ be a parallelogram with $AB = 5$, $BC = 3$, and $\angle BAD = 60^o$ . Let the angle bisector of $\angle ADC$ meet $AC$ at $E$ and $AB$ at $F$. The length $EF$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]B12.[/b] Find the sum of all positive integers $n$ such that $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$. Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [u]Set 5[/u] [b]B13.[/b] This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? [b]B14.[/b] Let $f(x) = \frac{1}{x} - 1$. Evaluate $$f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .$$ [b]B15.[/b] Square $WXYZ$ is inscribed in square $ABCD$ with side length $1$ such that $W$ is on $AB$, $X$ is on $BC$, $Y$ is on $CD$, and $Z$ is on $DA$. Line $W Y$ hits $AD$ and $BC$ at points $P$ and $R$ respectively, and line $XZ$ hits $AB$ and $CD$ at points $Q$ and $S$ respectively. If the area of $WXYZ$ is $\frac{13}{18}$ , then the area of $PQRS$ can be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. What is $m + n$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.

Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?