If $x$ is such that $\dfrac{1}{x}<2$ and $\dfrac{1}{x}>-3$, then: $\textbf{(A)}\ -\dfrac{1}{3}<x<\dfrac{1}{2} \qquad \textbf{(B)}\ -\dfrac{1}{2}<x<3 \qquad \textbf{(C)}\ x>\dfrac{1}{2} \qquad\\ \textbf{(D)}\ x>\dfrac{1}{2}\text{ or }-\dfrac{1}{3}<x<0 \qquad \textbf{(E)}\ x>\dfrac{1}{2}\text{ or }x<-\dfrac{1}{3}$

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(A,+,\cdot)$ be an unit ring with the property: for all $x\in A$, \[ x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} \] [list=a] [*]Let $x\in A$ and let $n\geq2$ be an integer such that $x^{n}=0$. Prove that $x=0$. [*]Prove that $x^{4}=x$, for all $x\in A$.[/list]

Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.

Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

A convex $n-$gon is split into three convex polygons. One of them has $n$ sides, the second one has more than $n$ sides, the third one has less than $n$ sides. Find all possible values of $n.$

Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.

[asy] unitsize(15); for (int a=0; a<6; ++a) { draw(2*dir(60a)--2*dir(60a+60),linewidth(1)); } draw((1,1.7320508075688772935274463415059)--(1,3.7320508075688772935274463415059)--(-1,3.7320508075688772935274463415059)--(-1,1.7320508075688772935274463415059)--cycle,linewidth(1)); fill((.4,1.7320508075688772935274463415059)--(0,3.35)--(-.4,1.7320508075688772935274463415059)--cycle,black); label("1.",(0,-2),S); draw(arc((1,1.7320508075688772935274463415059),1,90,300,CW)); draw((1.5,0.86602540378443864676372317075294)--(1.75,1.7)); draw((1.5,0.86602540378443864676372317075294)--(2.2,1)); draw((7,0)--(6,1.7320508075688772935274463415059)--(4,1.7320508075688772935274463415059)--(3,0)--(4,-1.7320508075688772935274463415059)--(6,-1.7320508075688772935274463415059)--cycle,linewidth(1)); draw((7,0)--(6,1.7320508075688772935274463415059)--(7.7320508075688772935274463415059,2.7320508075688772935274463415059)--(8.7320508075688772935274463415059,1)--cycle,linewidth(1)); label("2.",(5,-2),S); draw(arc((7,0),1,30,240,CW)); draw((6.5,-0.86602540378443864676372317075294)--(7.1,-.7)); draw((6.5,-0.86602540378443864676372317075294)--(6.8,-1.5)); draw((14,0)--(13,1.7320508075688772935274463415059)--(11,1.7320508075688772935274463415059)--(10,0)--(11,-1.7320508075688772935274463415059)--(13,-1.7320508075688772935274463415059)--cycle,linewidth(1)); draw((14,0)--(13,-1.7320508075688772935274463415059)--(14.7320508075688772935274463415059,-2.7320508075688772935274463415059)--(15.7320508075688772935274463415059,-1)--cycle,linewidth(1)); label("3.",(12,-2.5),S); draw((21,0)--(20,1.7320508075688772935274463415059)--(18,1.7320508075688772935274463415059)--(17,0)--(18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--cycle,linewidth(1)); draw((18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--(20,-3.7320508075688772935274463415059)--(18,-3.7320508075688772935274463415059)--cycle,linewidth(1)); label("4.",(19,-4),S);[/asy] The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4$? [asy] unitsize(12); label("(A)",(0,0),W); fill((1,-1)--(1,1)--(5,0)--cycle,black); label("(B)",(6,0),E); fill((9,-2)--(11,-2)--(10,1)--cycle,black); label("(C)",(14,0),E); fill((17,1)--(19,1)--(18,-1.8)--cycle,black); label("(D)",(22,0),E); fill((25,-1)--(27,-2)--(28,1)--cycle,black); label("(E)",(31,0),E); fill((33,0)--(37,1)--(37,-1)--cycle,black);[/asy]

For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have?

Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation \[ y = ax^2 + bx + c, \] and the lines defined by the six equations \begin{align*} &y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\ &y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.

On the sides of the regular $ n $-gon, $ n + 1 $ points are taken dividing the perimeter into equal parts. At what position of the selected points is the area of the convex polygon with these $ n + 1 $ vertices a) the largest, b) the smallest?

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Given two conditions: A: $\sqrt{1+\sin\theta}=a$ B: $\sin\frac{\theta}{2}+\cos\frac{\theta}{2}=a$ Then, which one of the followings are true? $(\text{A})$A is sufficient and necessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient but unnecessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Find the remainder when $2024^{2023^{2022^{2021...^{3^{2}}}}} + 2025^{2021^{2017^{2013...^{5^{1}}}}}$ is divided by $19$.

The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/d90827028f426f2d2772f7d7b875eea4909211.png[/img]

Consider the $5\times 5$ grid $Z^2_5 = \{(a, b) : 0 \le a, b \le 4\}$. Say that two points $(a, b)$,$(x, y)$ are adjacent if $a - x \equiv -1, 0, 1$ (mod $5$) and $b - y \equiv -1, 0, 1$ (mod $5$) . For example, in the diagram, all of the squares marked with $\cdot$ are adjacent to the square marked with $\times$. [img]https://cdn.artofproblemsolving.com/attachments/2/6/c49dd26ab48ddff5e1beecfbd167d5bb9fe16d.png[/img] What is the largest number of $\times$ that can be placed on the grid such that no two are adjacent?

On the table there are $2n$ coins that look the same. It is known that $n$ of them weigh 9 g. each, while the remaining $n$ weigh 10 g. each. It is required to split the coins into $n$ pairs with total weight of each pair 19 g. Prove that this can be done in less than $n$ weighings using a balance without additional weights (the balance shows which pan is heavier or that their weight is equal).

On a $ \$10000$ order a merchant has a choice between three successive discounts of $ 20 \%$, $ 20 \%$, and $ 10\%$ and three successive discounts of $ 40 \%$, $ 5 \%$, and $ 5 \%$. By choosing the better offer, he can save: $ \textbf{(A)}\ \text{nothing at all} \qquad \textbf{(B)}\ \$440 \qquad \textbf{(C)}\ \$330 \qquad \textbf{(D)}\ \$345 \qquad \textbf{(E)}\ \$360$

An acute triangle $ ABC$ with $ AB \neq AC$ is given. Let $ V$ and $ D$ be the feet of the altitude and angle bisector from $ A$, and let $ E$ and $ F$ be the intersection points of the circumcircle of $ \triangle AVD$ with sides $ AC$ and $ AB$, respectively. Prove that $ AD$, $ BE$ and $ CF$ have a common point.

What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$? $ \textbf{(A)}\ 49 \qquad\textbf{(B)}\ 53 \qquad\textbf{(C)}\ 97 \qquad\textbf{(D)}\ 103 \qquad\textbf{(E)}\ \text{None of the above} $

Peter picked a positive integer, multiplied it by 5, multiplied the result by 5,then multiplied the result by 5 again and so on. Altogether $k$ multiplications were made. It so happened that the decimal representations of the original number and of all $k$ resulting numbers in this sequence do not contain digit $7$. Prove that there exists a positive integer such that it can be multiplied $k$ times by $2$ so that no number in this sequence contains digit $7$.

On a circle there are $n$ points such that every point has a distinct number. Determine the number of ways of choosing $k$ points such that for any point there are at least 3 points between this point and the nearest point. (clockwise) ($n,k\geq 2$)

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]