For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6) = 2$ and $b(23)=5$. If $S = \textstyle\sum_{p=1}^{2007}b(p)$, find the remainder when S is divided by 1000.

The real numbers $x_1,\ldots ,x_{2011}$ satisfy \[x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'\] where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$. Prove that $x_1=x_2=\ldots =x_{2011}$ .

Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n\plus{}1}(a) \equal{} 2 \minus{} \frac{a}{f_n(a)}$, $ f_1(a) \equal{} 2$, $ n\equal{}1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n\plus{}k}(a) \equal{} f_{k}(a), k\equal{}1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$. Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a\minus{}1)\equal{}0$, where $ \displaystyle p_n(x)\equal{}\sum_{k\equal{}0}^{\left[ \frac{n\minus{}1}{2} \right]} (\minus{}1)^k C_n^{2k\plus{}1}x^k$, here we define $ \displaystyle \frac{a}{0}\equal{} \infty$ and $ \displaystyle \frac{a}{\infty} \equal{} 0$.

Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$. Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.

There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board, erases it, then writes all divisors of $a$ except $a$( Can be same numbers on the board). After some time on the board there are $N^2$ numbers. For which $N$ is it possible?

Let $\mathcal{S}$ be the set of all nonconstant polynomials $P$ with integer coefficients satisfying $P(\sqrt{3}+\sqrt{2})=P(\sqrt{3}-\sqrt{2}).$ If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0).$

Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation $$f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}$$ for all real numbers $x$ and $y$.

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$ a) for all $n\ge 2$ b) for all $n \ge 3$ (I. Gorodnin)

Given an integer $c$, the sequence $a_0, a_1, a_2, ...$ is generated using the recurrence relation $a_0 = c$ and $a_i = a^i_{i-1} + 2021a_{i-1}$ for all $i \ge 1$. Given that $a_0 = c$, let $f(c)$ be the smallest positive integer $n$ such that $a_n - 1$ is a multiple of $47$. Compute $$\sum^{46}_{k=1} f(k).$$

The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? [i]Proposed by Walther Janous [/i]

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1 + ax_2 + x_3 = b $$ $$x_2 + ax_3 + x_4 = b $$ $$x_3 + ax_4 + x_1 = b $$ $$x_4 + ax_1 + x_2 = b$$ Here $a$ and $b$ are real numbers (case distinction!).

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$.

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.

[u]Round 1[/u] [b]p1. [/b] What is the sum of the digits in the binary representation of $2023$? [b]p2.[/b] Jack is buying fruits at the EMCCmart. Three apples and two bananas cost $\$11.00$. Five apples and four bananas cost $\$19.00$. In cents, how much more does an apple cost than a banana? [b]p3.[/b] Define $a \sim b$ as $a! - ab$. What is $(4 \sim 5) \sim (5 \sim (3 \sim 1))$? [u] Round 2[/u] [b]p4.[/b] Alan has $24$ socks in his drawer. Of these socks, $4$ are red, $8$ are blue, and $12$ are green. Alan takes out socks one at a time from his drawer at random. What is the minimum number of socks he must pull out to guarantee that the number of green socks is at least twice the number of red socks? [b]p5.[/b] What is the remainder when the square of the $24$th smallest prime number is divided by $24$? [b]p6.[/b] A cube and a sphere have the same volume. If $k$ is the ratio of the length of the longest diagonal of the cube to the diameter of the sphere, find $k^6$. [u]Round 3[/u] [b]p7.[/b] Equilateral triangle $ABC$ has side length $3\sqrt3$. Point $D$ is drawn such that $BD$ is tangent to the circumcircle of triangle $ABC$ and $BD = 4$. Find the distance from the circumcenter of triangle $ABC$ to $D$. [b]p8.[/b] If $\frac{2023!}{2^k}$ is an odd integer for an integer $k$, what is the value of $k$? [b]p9.[/b] Let $S$ be a set of 6 distinct positive integers. If the sum of the three smallest elements of $S$ is $8$, and the sum of the three largest elements of $S$ is $19$, find the product of the elements in $S$. [u]Round 4[/u] [b]p10.[/b] For some integers $b$, the number $1 + 2b + 3b^2 + 4b^3 + 5b^4$ is divisible by $b + 1$. Find the largest possible value of $b$. [b]p11.[/b] Let $a, b, c$ be the roots of cubic equation $x^3 + 7x^2 + 8x + 1$. Find $a^2 + b^2 + c^2 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ [b]p12.[/b] Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3131590p28370327]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime number and the decimal notation of $\frac{1}{p}$ is periodical with a length of the period $4k$, $\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}…$ .Prove that $a_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}$.