Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$. [i]Proposed by Ahaan S. Rungta[/i]

Let $S=\{1, 2, 3, \dots 2021\}$ and $f:S \to S$ be a function such that $f^{(n)}(n)=n$ for each $n \in S$. Find all possible values for $f(2021)$. (Here, $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$.) [i]Authored by Viktor Simjanoski[/i]

A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.

Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$. Prove that $cx^2-bx + a > cx-b$ for any $x$.

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point ${z}$, satisfy $$\frac {1+z^{23}}{z^{64}}\in\mathbb R.$$ (2) For any non-zero complex number ${a}$ and any real number $\theta$, the equation $1+z^{23}+az^{64}=0$ has roots in $$S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.$$ [i]Proposed by Yijun Yao[/i]

Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

Find all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

Let $a_0,a_1,a_2 \dots a_n, \dots$ be an infinite sequence of real numbers, defined by $$a_0 = c$$ $$a_{n+1} = {a_n}^2+\frac{a_n}{2}+c$$ for some real $c > 0$. Find all values of $c$ for which the sequence converges and the limit for those values.

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

Find all real numbers $x$ such that: \[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]

[b]p1.[/b] Find the number of zeroes at the end of $20^{17}$. [b]p2.[/b] Express $\frac{1}{\sqrt{20} +\sqrt{17}}$ in simplest radical form. [b]p3.[/b] John draws a square $ABCD$. On side $AB$ he draws point $P$ so that $\frac{BP}{PA}=\frac{1}{20}$ and on side $BC$ he draws point $Q$ such that $\frac{BQ}{QC}=\frac{1}{17}$ . What is the ratio of the area of $\vartriangle PBQ$ to the area of $ABCD$? [b]p4.[/b] Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints? [b]p5.[/b] Alex is playing a game with an unfair coin which has a $\frac15$ chance of flipping heads and a $\frac45$ chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins? [b]p6.[/b] Positive two-digit number $\overline{ab}$ has $8$ divisors. Find the number of divisors of the four-digit number $\overline{abab}$. [b]p7.[/b] Call a positive integer $n$ diagonal if the number of diagonals of a convex $n$-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to $2017$. [b]p8.[/b] There are $4$ houses on a street, with $2$ on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors. [b]p9.[/b] Compute $$|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.$$ [b]p10.[/b] Given points $A,B$ in the coordinate plane, let $A \oplus B$ be the unique point $C$ such that $\overline{AC}$ is parallel to the $x$-axis and $\overline{BC}$ is parallel to the $y$-axis. Find the point $(x, y)$ such that $((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y)$. [b]p11.[/b] In the following subtraction problem, different letters represent different nonzero digits. $\begin{tabular}{ccccc} & M & A & T & H \\ - & & H & A & M \\ \hline & & L & M & T \\ \end{tabular}$ How many ways can the letters be assigned values to satisfy the subtraction problem? [b]p12.[/b] If $m$ and $n$ are integers such that $17n +20m = 2017$, then what is the minimum possible value of $|m-n|$? [b]p13. [/b]Let $f(x)=x^4-3x^3+2x^2+7x-9$. For some complex numbers $a,b,c,d$, it is true that $f (x) = (x^2+ax+b)(x^2+cx +d)$ for all complex numbers $x$. Find $\frac{a}{b}+ \frac{c}{d}$. [b]p14.[/b] A positive integer is called an imposter if it can be expressed in the form $2^a +2^b$ where $a,b$ are non-negative integers and $a \ne b$. How many almost positive integers less than $2017$ are imposters? [b]p15.[/b] Evaluate the infinite sum $$\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...$$ [b]p16.[/b] Each face of a regular tetrahedron is colored either red, green, or blue, each with probability $\frac13$ . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors? [b]p17.[/b] Let $(k,\sqrt{k})$ be the point on the graph of $y=\sqrt{x}$ that is closest to the point $(2017,0)$. Find $k$. [b]p18.[/b] Alice is going to place $2016$ rooks on a $2016 \times 2016$ chessboard where both the rows and columns are labelled $1$ to $2016$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations. [b]p19.[/b] Let $f (n)$ be a function defined recursively across the natural numbers such that $f (1) = 1$ and $f (n) = n^{f (n-1)}$. Find the sum of all positive divisors less than or equal to $15$ of the number $f (7)-1$. [b]p20.[/b] Find the number of ordered pairs of positive integers $(m,n)$ that satisfy $$gcd \,(m,n)+ lcm \,(m,n) = 2017.$$ [b]p21.[/b] Let $\vartriangle ABC$ be a triangle. Let $M$ be the midpoint of $AB$ and let $P$ be the projection of $A$ onto $BC$. If $AB = 20$, and $BC = MC = 17$, compute $BP$. [b]p22.[/b] For positive integers $n$, define the odd parent function, denoted $op(n)$, to be the greatest positive odd divisor of $n$. For example, $op(4) = 1$, $op(5) = 5$, and $op(6) =3$. Find $\sum^{256}_{i=1}op(i).$ [b]p23.[/b] Suppose $\vartriangle ABC$ has sidelengths $AB = 20$ and $AC = 17$. Let $X$ be a point inside $\vartriangle ABC$ such that $BX \perp CX$ and $AX \perp BC$. If $|BX^4 -CX^4|= 2017$, the compute the length of side $BC$. [b]p24.[/b] How many ways can some squares be colored black in a $6 \times 6$ grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct. [b]p25.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = 2$, $AD = 4$, and $\angle ABC = 120^o$. Let $M$ be the midpoint of $BD$. If $\angle AMC = 90^o$, find the length of segment $CD$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is owing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

[b]p1.[/b] Find all real solutions of the system $$x^5 + y^5 = 1$$ $$x^6 + y^6 = 1$$ [b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the circles, which are outside of the triangle. [b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits. [b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves? [b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer. [b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer. PS. You should use hide for answers.

Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$ $$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$

A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.

Solve the system of equations $$\begin{cases} x+y+z+t=6 \\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8 \end{cases}$$

Alina knows how to twist a periodic decimal fraction in the following way: she finds the minimum preperiod of the fraction, then takes the number that makes up the period and rearranges the last one in it digit to the beginning of the number. For example, from the fraction, $0.123(56708)$ she will get $0.123(85670)$. What fraction will Alina get from fraction $\frac{503}{2022}$ ?

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.

Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$