For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{} m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\] [i]V. T. Sos[/i]

Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side $n$. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2). Prove that if $n \ge 3$ the second arrangement cannot be obtained from the first one by moving piece Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)

In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$ [asy] unitsize(75); pathpen = black; pointpen=black; pair A = MP("A", D((0,0)), dir(200)); pair B = MP("B", D((2,0)), dir(-20)); pair C = MP("C", D((1/2,1)), dir(100)); pair D = MP("D", D(midpoint(B--C)), dir(30)); pair E = MP("E", D(midpoint(A--B)), dir(-90)); pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013); draw(A--B--C--cycle); draw(A--D--E--C); [/asy] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 13.5 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 14.5 \qquad \textbf{(E)}\ 15 $

Students from three middle schools worked on a summer project. *Seven students from Allen school worked for $3$ days. *Four students from Balboa school worked for $5$ days. *Five students from Carver school worked for $9$ days. The total amount paid for the students' work was $ \$774$. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether? $\text{(A)}\ 9.00\text{ dollars} \qquad \text{(B)}\ 48.38\text{ dollars} \qquad \text{(C)}\ 180.00\text{ dollars} \qquad \text{(D)}\ 193.50\text{ dollars} \qquad \text{(E)}\ 258.00\text{ dollars}$

Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold? [i]Dorel Miheț[/i]

Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

Let $ ABCD $ be a sqare and $ E $ be a point situated on the segment $ BD, $ but not on the mid. Denote by $ H $ and $ K $ the orthocenters of $ ABE, $ respectively, $ ADE. $ Show that $ \overrightarrow{BH}=\overrightarrow{KD} . $

We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$, in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards). Prove that starting from the point $\left(1;\sqrt 2\right)$, we cannot return to it in finitely many steps.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

What is the units digit of $2022^{2023} + 2023^{2022}$? $\textbf{(A)}~7\qquad\textbf{(B)}~1\qquad\textbf{(C)}~3\qquad\textbf{(D)}~5\qquad\textbf{(E)}~9$

Select $ n$ points on a circle independently with uniform distribution. Let $ P_n$ be the probability that the center of the circle is in the interior of the convex hull of these $ n$ points. Calculate the probabilities $ P_3$ and $ P_4$. [A. Renyi]

At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the [i]product[/i] of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are $8^6=262144$ participants, and no two get the same score on every problem. Find the score of the participant whose rank was $7^6 = 117649$. [i]Proposed by Yang Liu[/i]

What is the least positive integer with the property that the product of its digits is $9! ?$

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$ $$f(y)f(x+f(y))=f(x)f(xy)$$

Let $p$ be a prime with $p \equiv 1 \pmod{4}$. Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol.

Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.

A positive integer $N$ greater than $1$ is described as special if in its base-$8$ and base-$9$ representations, both the leading and ending digit of $N$ are equal to $1$. What is the smallest special integer in decimal representation? [i]Proposed by Michael Ren[/i]

Given a set $S$ of integers, an allowed operation consists of the following three steps: $\bullet$ Choose a positive integer $n$. $\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct. $\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$. Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$. Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.

The following square table is given with seven raws and seven columns: $a_{11},a_{12},a_{13},a_{14},a_{15},a_{16},a_{17}$ $a_{21},a_{22},a_{23},a_{24},a_{25},a_{26},a_{27}$ $a_{31},a_{32},a_{33},a_{34},a_{35},a_{36},a_{37}$ $a_{41},a_{42},a_{43},a_{44},a_{45},a_{46},a_{47}$ $a_{51},a_{52},a_{53},a_{54},a_{55},a_{56},a_{57}$ $a_{61},a_{62},a_{63},a_{64},a_{65},a_{66},a_{67}$ $a_{71},a_{72},a_{73},a_{74},a_{75},a_{76},a_{77}$ Suppose $a_{ij}\in\{0,1\},\forall i,j\in\{1,...,7\}$. Prove that there exists at least one combination of the numbers $1$ and $0$ so that the following conditions hold: $(i)$ Each raw and each column has exactly three $1$'s. $(ii)$$\sum_{j=1}^7a_{lj}a_{ij}=1,\forall l,i\in\{1,...,7\}$ and $l\neq i$.(so for any two distinct raws there is exactly one $r$ so that the both raws have $1$ in the $r$-th place). $(iii)$$\sum_{i=1}^7a_{ij}a_{ik}=1,\forall j,k\in\{1,...,7\}$ and $j\neq k$.(so for any two distinct columns there is exactly one $s$ so that the both columns have $1$ in the $s$-th place).

Let $ ABC$ be a triangle. Circle $ \omega$­ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$­ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of ­$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.

We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum. (The total distance of the point P from the sides of an equilateral triangle is fixed ) [i]Proposed by Erfan Salavati[/i]

The incircle $\omega$ of a triangle $ABC$ touches $BC, AC$ and $AB$ at points $A_0, B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to segment $AA_0$ at points $Q$ and $P$ respectively. Prove that $PC_0$ and $QB_0$ meet on $\omega$ .

Source: 2017 Canadian Open Math Challenge, Problem C3 ----- Let $XYZ$ be an acute-angled triangle. Let $s$ be the side-length of the square which has two adjacent vertices on side $YZ$, one vertex on side $XY$ and one vertex on side $XZ$. Let $h$ be the distance from $X$ to the side $YZ$ and let $b$ be the distance from $Y$ to $Z$. [asy] pair S, D; D = 1.27; S = 2.55; draw((2, 4)--(0, 0)--(7, 0)--cycle); draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle); label("$X$",(2,4),N); label("$Y$",(0,0),W); label("$Z$",(7,0),E); [/asy] (a) If the vertices have coordinates $X = (2, 4)$, $Y = (0, 0)$ and $Z = (4, 0)$, find $b$, $h$ and $s$. (b) Given the height $h = 3$ and $s = 2$, find the base $b$. (c) If the area of the square is $2017$, determine the minimum area of triangle $XYZ$.

Defined all $f : \mathbb{R} \to \mathbb{R} $ that satisfied equation $$f(x)f(y)f(x-y)=x^2f(y)-y^2f(x)$$ for all $x,y \in \mathbb{R}$