Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle. Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of the lamp; if $i$ is turned on, a [i]signal[/i] of range $i$ is sent by this lamp, and it will be received only by the next $i$ lamps which follow $i$, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed. The example in attachment, for $n=4$, ilustrates a configuration where lamps $2$ and $4$ are initially turned on. Lamp $2$ sends signal only for the lamps $3$ e $4$, while lamp $4$ sends signal for lamps $1$, $2$, $3$ e $4$. Therefore, we verify that lamps $1$ e $2$ received only one signal, while lamps $3$ e $4$ received two signals. Therefore, in the next configuration, lamps $1$ e $4$ will be turned on, while lamps $2$ e $3$ will be turned off. Let $\Psi$ to be the set of all $2^n$ possible configurations, where $0 \le \ell \le n$ random lamps are turned on. We define a function $f: \Psi \rightarrow \Psi$ where, if $\xi$ is a configuration of lamps, then $f(\xi)$ is the configurations obtained after we perform the [i]cool procedure[/i] described above. Determine all values of $n$ for which $f$ is bijective.

Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.

Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.

Determine all functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|wf(z)+zf(w)|=2|zw|$$ for all $w,z\in\mathbb{C}$.

We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that: $i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$ $ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$ $iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$ Find all [i]interesting [/i]$n$. [i]Proposed by Mojtaba Zare Bidaki[/i]

Suppose $a$ is a non-zero real number such that $a +\frac{1}{a}$ is a whole number. (a) Prove that $a^2 +\frac{1}{a^2}$ is also an integer. (b) Prove that $a^n+\frac{1}{a^n}$ is also an integer, for any integer value positive of $n$.

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

Let $S = \{-17, -16, ..., 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for all $x \in T$ and if $x, y, z \in T (x, y, z$ may be equal) then $x + y + z \ne 0$. Find the largest number of elements in a good set.

p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$. Find the number of subsets of $H$. p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$. a. Give an example of your own from the above process. b. Explain mathematically the general form of the process. p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn? p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made. p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Let $n,k$ be given natural numbers. Determine all ordered n-tuples of non-negative real numbers $(x_1,x_2,...,x_n)$ that satisfy the system of equations $$x_1^k+x_2^k+...+x_n^k=1$$ $$(1+x_1)(1+x_2)...(1+x_n)=2$$

If $x$ and $y$ are positive real numbers such that $\sqrt(2x)+\sqrt(y)=13$ and $\sqrt(8x)+\sqrt(9y)=35$, calculate $20x+23y$.

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

A mushroom is called [i]bad [/i] if it contains at least $10$ worms. A basket contains $90$ bad and $10$ good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones? [hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?[/hide]

Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]

Let $x, y$ be numbers in the interval (0,1) such that for some $a>0, a \neq 1$ $$\log _{x} a+\log _{y} a=4 \log _{x y} a$$Prove that $x=y$

[u]Round 5[/u] [b]p13.[/b] Mandy is baking cookies. Her recipe calls for $N$ grams of flour, where $N$ is the number of perfect square divisors of $20! + 24!$. Find $N$. [b]p14.[/b] Consider a circular table with center $R$. Beef-loving Bryan places a steak at point $I$ on the circumference of the table. Then he places a bowl of rice at points $C$ and $E$ on the circumference of the table such that $CE \parallel IR$ and $\angle ICE = 25^o$. Find $\angle CIE$. [b]p15.[/b] Enya writes the $4$-letter words $LEEK$, $BEAN$, $SOUP$, $PEAS$, $HAMS$, and $TACO$ on the board. She then thinks of one of these words and gives Daria, Ava, Harini, and Tiffany a slip of paper containing exactly one letter from that word such that if they ordered the letters on their slips correctly, they would form the word. Each person announces at the same time whether they know the word or not. Ava, Harini, and Tiffany all say they do not know the word, while Daria says she knows the word. After hearing this, Ava, Harini, and Tiffany all know the word. Assuming all four girls are perfect logicians and they all thought of the same correct word, determine Daria’s letter. [u]Round 6[/u] [b]p16.[/b] Michael receives a cheese cube and a chocolate octahedron for his 5th birthday. On every day after, he slices off each corner of his cheese and chocolate with a knife. Each slice cuts off exactly one corner. He then eats each corner sliced off. Find the difference between the total number of cheese and chocolate pieces he has eaten by the end of his $6$th birthday. (Michael’s $5$th and $6$th birthdays do not occur on leap years.) [b]p17.[/b] Let $D$ be the average of all positive integers n satisfying $$lcm (gcd (n, 2000), gcd (n, 24)) = gcd (lcm (n, 2000), lcm (n, 24)).$$ Find $3D$. [b]p18.[/b] The base $\vartriangle ABC$ of the triangular pyramid $PABC$ is an equilateral triangle with a side length of $3$. Given that $PA = 3$, $PB = 4$, and $PC = 5$, find the circumradius of $PABC$. [u]Round 7[/u] [b]p19.[/b] $2049300$ points are arranged in an equilateral triangle point grid, a smaller version of which is shown below, such that the sides contain $2024$ points each. Peter starts at the topmost point of the grid. At $9:00$ am each day, he moves to an adjacent point in the row below him. Derrick wants to prevent Peter from reaching the bottom row, so at $12:00$ pm each day, he selects a point on the bottom row and places a rock at that point. Peter stops moving as soon as he is guaranteed to end up at a point with a rock on it. At least how many moves will Peter complete, no matter how Derrick places the rocks? [img]https://cdn.artofproblemsolving.com/attachments/f/a/346d25a5d7bb7a5fbefae7edad727965312b25.png[/img] [b]p20.[/b] There are $N$ stones in a pile, where $N$ is a positive integer. Ava and Anika take turns playing a game, with Ava moving first. If there are n stones in the pile, a move consists of removing $x$ stones, where $1 < gcd(x, n) \le x < n$. Whoever first has no possible moves on their turn wins. Both Ava and Anika play optimally. Find the $2024$th smallest value of $N$ for which Ava wins. [b]p21.[/b] Alan is bored and alone, so he plays a fun game with himself. He writes down all quadratic polynomials with leading coefficient $1$ whose coefficients are integers between $-10$ and $10$, inclusive, on a blackboard. He then erases all polynomials which have a non-integer root. Alan defines the size of a polynomial $P(x)$ to be $P(1)$ and spends an hour adding up the sizes of all the polynomials remaining on the blackboard. Assuming Alan does computation perfectly, find the sum Alan obtains. [u]Round 8[/u] [b]p22.[/b] A prime number is a positive integer with exactly two distinct divisors. You must submit a prime number for this problem. If you do not submit a prime number, you gain $0$ points, and your submission will not be considered valid. The median of all valid submitted numbers is $M$ (duplicates are counted). Estimate $2M$. If your team’s absolute difference between $2M$ and your submission is the $i$th smallest absolute difference among all teams, you gain max$(23 - 2i, 0)$ points. All teams who did not submit any number gain $0$ points. (In the case of a tie, all teams that tied gain the same amount of points.) [b]p23.[/b] Ribbotson the Frog is at the point $(0, 0)$ and wants to reach the point $(18, 18)$ in $36$ steps. Each step, he either moves one unit in the $+x$ direction or one unit in the $+y$ direction. However, Ribbotson hates turning, so he must make at least two steps in any direction before switching directions. If $m$ is the number of different paths Ribbotson the Frog can make, estimate $m$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21\left(1 -\left|\log_{10}\frac{N}{m} \right|^2\right)$. [b]p24.[/b] Let $M = \pi^{\pi^{\pi^{\pi}}}$. Estimate $k$, where $M = 10^{10^{k}}$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21 \cdot 1.01^{(-|N-k|^3)}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3248729p29808138]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by $$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$ Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.

Is it true that for every $n\ge 2021$ there exist $n$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.