On a board are written some positive reals (not necessarily distinct). For every two numbers in the frame $a$ and $b$ distinct such that $$\frac{1}{2}<\frac{a}{b}<2,$$ an allowed operation is to delete $a$ and $b$ and write $2a-b$ and $2b-a$ in their place. Show that we can do the operation only a finite number of times.

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.

Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?

Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer. [i]A. Golovanov[/i]

Let $n$ be a positive integer. Find the value of the following sum \[\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},\] where $e_k\in\{0,1\}$ for $1\leq k \leq n$, and the sum $\sum_{(n)}$ is taken over all $2^n$ possible choices of $e_1,\ldots ,e_n$.

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$. Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions: 1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied. 2) For every pair of real numbers $x$ and $y$, \[ f(xf(y))+yf(x)=xf(y)+f(xy)\] is satisfied.

[b]p1.[/b] On the island of Nevermind some people are liars; they always lie. The remaining habitants of the island are truthlovers; they tell only the truth. Three habitants of the island, $A, B$, and $C$ met this morning. $A$ said: “All of us are liars”. $B$ said: “Only one of us is a truthlover”. Who of them is a liar and who of them is a truthlover? [b]p2.[/b] Pinocchio has $9$ pieces of paper. He is allowed to take a piece of paper and cut it in $5$ pieces or $7$ pieces which increases the number of his pieces. Then he can take again one of his pieces of paper and cut it in $5$ pieces or $7$ pieces. He can do this again and again as many times as he wishes. Can he get $2004$ pieces of paper? [b]p3.[/b] In Dragonland there are coins of $1$ cent, $2$ cents, $10$ cents, $20$ cents, and $50$ cents. What is the largest amount of money one can have in coins, yet still not be able to make exactly $1$ dollar? [b]p4.[/b] Find all solutions $a, b, c, d, e$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & d \\ + & a & c & a & c \\ \hline c & d & e & b & c \\ \end{tabular}$ [b]p5.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Find the last digit of $2022^{2022}$. [b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$. [b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$. [b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$. [b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$. [b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$. [b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. [b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?

a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate? b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

Let $ P[X,Y] $ be a polynomial of degree at most $ 2 .$ If $ A,B,C,A',B',C' $ are distinct roots of $ P $ such that $ A,B,C $ are not collinear and $ A',B',C' $ lie on the lines $ BC,CA, $ respectively, $ AB, $ in the planar representation of these points, show that $ P=0. $

Prove that there are no functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x,y\in \mathbb{R}:$ $ f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y$ and $f(x)\geq x^2$. [i]Proposed by Mohammad Ahmadi[/i]

Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true: \begin{align*} f(a)&=f(b), <br /> \\ f(a+b)&=\min\{f(a),f(b)\}. \end{align*} [i]Remarks:[/i] $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\to Y$ is said to be surjective if for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$.

In a race, people rode either bicycles with blue wheels or tricycles with tan wheels. Given that 15 more people rode bicycles than tricycles and there were 15 more tan wheels than blue wheels, what is the total number of people who rode in the race?

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Determine all real numbers $x{}$ satisfying $2^{x-1}+2^{1/\sqrt{x}}=3$.

Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

A traffic light installed at a main junction of a road, in which you circulate in both directions, it remains red for $30$ s and green for another $30$ s, alternately. You want to install another traffic light on the same road, for a secondary crossing, located $400$ m away from the first, which works with the same period of $1$ min duration. It is wanted that the cars that circulate at $60$ km/h on the road in any of the two senses and that they do not have to stop if there was only the traffic light of the main intersection. They also don't have to stop after installing the secondary crossover. How many seconds can red be on at the secondary traffic light? Note: It is suggested to reason on a Cartesian representation of the march of the cars, taking an axis of distances and another of times.