Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

Assume that $a$ and $b$ are natural numbers with $a \ge b$ so that $ \sqrt{a+\sqrt{a^2-b^2}}$ is a natural number. Show that $a$ and $b$ have the same parity.

Let $\phi(k)$ denote the number of positive integers less than or equal to $k$ that are relatively prime to $k$. For example, $\phi(2)=1$ and $\phi(10)=4$. Compute the number of positive integers $n \leq 2020$ such that $\phi(n^2)=2\phi(n)^2$.

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC$, $DE$, and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m + n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$? $\textbf{(A)}~47\qquad\textbf{(B)}~52\qquad\textbf{(C)}~55\qquad\textbf{(D)}~58\qquad\textbf{(E)}~63$

Use the following description of a machine to solve the first 4 problems in the round. A machine displays four digits: $0000$. There are two buttons: button $A$ moves all digits one position to the left and fills the rightmost position with $0$ (for example, it changes $1234$ to $2340$), and button $B$ adds $11$ to the current number, displaying only the last four digits if the sum is greater than $9999$ (for example, it changes $1234$ to $1245$, and changes $9998$ to $0009$). We can denote a sequence of moves by writing down the buttons pushed from left to right. A sequence of moves that outputs $2100$, for example, is $BABAA$. [b]p1[/b]. Give a sequence of $17$ or less moves so that the machine displays $2020$. [b]p2.[/b] Using the same machine, how many outputs are possible if you make at most three moves? [b]p3.[/b] Button $ B$ now adds n to the four digit display, while button $ A$ remains the same. For how many positive integers $n \le 20$ (including $11$) can every possible four-digit output be reached? [b]p4.[/b] Suppose the function of button $ A$ changes to: move all digits one position to the right and fill the leftmost position with $2$. Then, what is the minimum number of moves required for the machine to display $2020$, if it initially displays $0000$? [b]p5.[/b] In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle’s sides. If the area of the largest triangle is $64$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/6/f/fe17b6a6d0037163f0980a5a5297c1493cc5bb.png[/img] [b]p6.[/b] A bee flies $10\sqrt2$ meters in the direction $45^o$ clockwise of North (that is, in the NE direction). Then, the bee turns $135^o$ clockwise, and flies $20$ forward meters. It continues by turning $60^o$ counterclockwise, and flies forward $14$ meters. Finally, the bee turns $120^o$ clockwise and flies another $14$ meters forward before finally finding a flower to pollinate. How far is the bee from its starting location in meters? [b]p7.[/b] All the digits of a $15$-digit number are either $p$ or $c$. $p$ shows up $3$ more times than $c$ does, and the average of the digits is $c - p$. What is $p + c$? [b]p8.[/b] Let $m$ be the sum of the factors of $75$ (including $1$ and $75$ itself). What is the ones digit of $m^{75}$ ? [b]p9.[/b] John flips a coin twice. For each flip, if it lands tails, he does nothing. If it lands heads, he rolls a fair $4$-sided die with sides labeled 1 through $4$. Let $a/b$ be the probability of never rolling a $3$, in simplest terms. What is $a + b$? [b]p10.[/b] Let $\vartriangle ABC$ have coordinates $(0, 0)$, $(0, 3)$,$(18, 0)$. Find the number of integer coordinates interior (excluding the vertices and edges) of the triangle. [b]p11.[/b] What is the greatest integer $k$ such that $2^k$ divides the value $20! \times 20^{20}$? [b]p12.[/b] David has $n$ pennies, where $n$ is a natural number. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $4$ pennies left; is David spends all his money on cranberries, he will have $6$ pennies left. What is the second least possible amount of pennies that David can have? [b]p13.[/b] Elvin is currently at Hopperville which is $40$ miles from Waltimore and $50$ miles from Boshington DC. He takes a taxi back to Waltimore, but unfortunately the taxi gets lost. Elvin now finds himself at Kinsville, but he notices that he is still $40$ miles from Waltimore and $50$ miles from Boshington $DC$. If Waltimore and Boshington DC are $30$ miles apart, What is the maximum possible distance between Hopperville and Kinsville? [b]p14.[/b] After dinner, Rick asks his father for $1000$ scoops of ice cream as dessert. Rick’s father responds, “I will give you $2$ scoops of ice cream, plus $ 1$ additional scoop for every ordered pair $(a, b)$ of real numbers satisfying $\frac{1}{a + b}= \frac{1}{a}+ \frac{1}{b}$ you can find.” If Rick finds every solution to the equation, how many scoops of ice cream will he receive? [b]p15.[/b] Esther decides to hold a rock-paper-scissors tournament for the $56$ students at her school. As a rule, competitors must lose twice before they are eliminated. Each round, all remaining competitors are matched together in best-of-1 rock-paper-scissors duels. If there is an odd number of competitors in a round, one random competitor will not compete that round. What is the maximum number of matches needed to determine the rock-paper-scissors champion? [b]p16.[/b] $ABCD$ is a rectangle. $X$ is a point on $\overline{AD}$, $Y$ is a point on $\overline{AB}$, and $N$ is a point outside $ABCD$ such that $XYNC$ is also a rectangle and $YN$ intersects $\overline{BC}$ at its midpoint $M$. $ \angle BYM = 45^o$. If $MN = 5$, what is the sum of the areas of $ABCD$ and $XYNC$? [b]p17. [/b] Mr. Brown has $10$ identical chocolate donuts and $15$ identical glazed donuts. He knows that Amar wants $6$ donuts, Benny wants $9$ donuts, and Callie wants $9$ donuts. How many ways can he distribute out his $25$ donuts? [b]p18.[/b] When Eric gets on the bus home, he notices his $ 12$-hour watch reads $03: 30$, but it isn’t working as expected. The second hand makes a full rotation in $4$ seconds, then makes another in $8$ seconds, then another in $ 12$ seconds, and so on until it makes a full rotation in $60$ seconds. Then it repeats this process, and again makes a full rotation in $4$ second, then $8$ seconds, etc. Meanwhile, the minute hand and hour hand continue to function as if every full rotation of the second hand represents $60$ seconds. When Eric gets off the bus $75$ minutes later, his watch reads $AB: CD$. What is $A + B + C + D$? [b]p19.[/b] Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between $12: 00$ and $13: 15$, and will wait for Betty for $15$ minutes before he leaves. Betty will arrive at the airport between $12: 30$ and $13: 10$, and will wait for Alex for $10$ minutes before she leaves. The chance that they arrive at any time in their respective time intervals is equally likely. The probability that they will meet at the airport can be expressed as $a/b$ where $a/b$ is a fraction written in simplest form. What is $a + b$? [b]p20.[/b] Let there be $\vartriangle ABC$ such that $A = (0, 0)$, $B = (23, 0)$, $C = (a, b)$. Furthermore, $D$, the center of the circle that circumscribes $\vartriangle ABC$, lies on $\overline{AB}$. Let $\angle CDB = 150^o$. If the area of $\vartriangle ABC$ is $m/n$ where $m, n$ are in simplest integer form, find the value of $m \,\, \mod \,\,n$ (The remainder of $m$ divided by $n$). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.

Let $\|x\|_*=(|x|+|x-1|-1)/2$. Find all $f:\mathbb{N}\to\mathbb{N}$ such that \[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\] Here $f^{(0)}(x)=x$ and $f^{(n)}(x)=f(f^{(n-1)}(x))$ for all $n\in\mathbb{N}$. [i]Proposed by usjl[/i]

Let $n$ be a positive integer. We have $n$ boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called [i]solvable[/i] if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.

There is a $4 \times 4$ array of integers $A$, all initially equal to $0$. An operation may be performed on the array for any row or column such that every number in that row or column has $1$ added to it, and then is replaced with its remainder modulo $3$. Given a random $4 \times 4$ array of integers between $0$ and $2$ not identical to $A$, the probability that it can be reached through a series of operations on $A$ is $\frac{p}{q},$ where $p,q$ are relatively prime positive integers. Find $p$.

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which \[ P(P(P(n))) > 10^{12} \]

Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Show that there exists a positive integer $N$ such that for all integers $a>N$, there exists a contiguous substring of the decimal expansion of $a$, which is divisible by $2011$. Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.

In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules:\\ $1$. It must be in the interior of the island (none on borders).\\ $2$. No two airports can be at the exact same location.\\ $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle.\\ $4$. No three airports can be collinear.\\ Find the maximum value of $k$ for each $n$ [hide=Harder Version]Replace $1$ new and $2$ old with $1$ old and $2$ new.[/hide]

For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together? $\textbf{(A)}\ \$37.50 \qquad\textbf{(B)}\ \$50.00\qquad\textbf{(C)}\ \$87.50\qquad\textbf{(D)}\ \$90.00\qquad\textbf{(E)}\ \$92.50$

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that: $i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$. $ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$. a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist. b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.

a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution. b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as $$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.

Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$. If there is a real number $C$, such that \[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\] for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.

Show that the number$$4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right)$$ is an integer and determine it.

Let $S=\{(x,y)\mid -1\leq xy\leq 1\}$ be a subset of the real coordinate plane. If the smallest real number that is greater than or equal to the area of any triangle whose interior lies entirely in $S$ is $A$, compute the greatest integer not exceeding $1000A$. [i]Proposed by Yannick Yao[/i]

Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares. (A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.)

a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that: $|a-b|+|b-c|+|c-a|\le2\sqrt{2}$ b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of: $S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$