[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases: (a) $x,y,z$ are arbitrary non-zero real numbers; (b) $a<x,y,z<b (1<a^{3}<b)$. [b](R. 13)[/b]

Greater or less than one is the number $0.99999^{1.00001} \cdot 1.00001^{0.99999}$?

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties: (i) $f(x)\neq x$, (ii) $f(f(x))=x$, (iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$. Prove that $n\equiv 2 \pmod4$. (Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)

Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

A sequence of pairwise distinct positive integers is called averaging if each term after the first two is the average of the previous two terms. Let $M$ be the maximum possible number of terms in an averaging sequence in which every term is less than or equal to $2022$ and let $N$ be the number of such distinct sequences (every term less than or equal to $2022$) with exactly $M$ terms. What is $M+N?$ (Two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are said to be distinct if $a_i \neq b_i$ for some integer $1 \leq i \leq n$). [i]Proposed by Kyle Lee[/i]

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, what is the smallest distance that the grasshoper can be away from the point $2010$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 34 \qquad\textbf{(E)}\ 164 $

On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions: (A) Gives a candy to the student sitting on his left or to the student sitting on his right. (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right. At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions. (Two distributions are different if there is a student who has a different number of candy in each of these distributions.) （Forgive my poor English）

$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear. [i]Proposed by chengbilly[/i]

The solution of the equations \begin{align*} 2x-3y&=7 \\ 4x-6y &=20 \\ \end{align*} is: $ \textbf{(A)}\ x=18, y=12 \qquad \textbf{(B)}\ x=0, y=0 \qquad \textbf{(C)}\ \text{There is no solution} \\ \textbf{(D)}\ \text{There are an unlimited number of solutions} \qquad \textbf{(E)}\ x=8, y=5$

Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$. Find max value capacity(volume) of the pyramid $SABC$.

We are given an $18\times 18$ table, all of whose cells may be black or white. Initially all the cells are coloured white. We may perform the following operation: choose one column or one row and change the colour of all cells in this column or row. Is it possible by repeating the operation to obtain a table with exactly $16$ black cells?

For a sequence $a_1<a_2<\cdots<a_n$ of integers, a pair $(a_i,a_j)$ with $1\leq i<j\leq n$ is called [i]interesting[/i] if there exists a pair $(a_k,a_l)$ of integers with $1\leq k<l\leq n$ such that $$\frac{a_l-a_k}{a_j-a_i}=2.$$ For each $n\geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.

Given points $A, M, M_1$ and rational number $\lambda \neq -1$. Construct the triangle $ABC$, such that $M$ lies on $BC$ and $M_1$ lies on $B_1C_1$ ($B_1, C_1$ are the projections of $B, C$ on $AC, AB$ respectively), and $\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.$

Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.

Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$? $\textbf{(A) } 10 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

Let $a$ and $b$ be positive real numbers such that $ab(a-b)=1$. Which of the followings can $a^2+b^2$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 2\sqrt 2 \qquad\textbf{(D)}\ \sqrt {11} \qquad\textbf{(E)}\ \text{None of the preceding} $

In triangle $ABC$, $|AB| <|BC|$ holds. Point $I$ is the center of the circle inscribed in that triangle. Let $M$ be the midpoint of the side $AC$, and $N$ be the midpoint of the arc $AC$ of the circumcircle of that triangle containing point $B$. Prove that $\angle IMA = \angle INB$.

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

Let $a$ and $b$ be positive real numbers such that $a+b = 1$. Prove that $$\frac12 \le \frac{a^3+b^3}{a^2+b^2} \le 1$$