Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ [i]lucky[/i]. For which positive integers $n$, one can find a lucky $n$-tuple?

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

If $ \log_{10}{m} \equal{} b \minus{} \log_{10}{n}$, then $ m$= $\textbf{(A)}\ \dfrac{b}{n} \qquad \textbf{(B)}\ bn \qquad \textbf{(C)}\ 10^b n\qquad \textbf{(D)}\ b-10^n \qquad \textbf{(E)}\ \dfrac{10^b}{n}$

Let $a,b,c$ be positive integers such that $a^2+b^2+c^2$ is divisble by $a+b+c$. Prove that at least two of the numbers $a^3,b^3,c^3$ leave the same remainder by division through $a+b+c$.

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is: $\textbf{(A)}\ 38.5 \qquad \textbf{(B)}\ 37.5 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 36.5 \qquad \textbf{(E)}\ 36$

Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

Antonio and Beltran have impeccable logical reasoning, they put on a hat with a integer between $0$ and $19$ (including both) so that each of them sees the number that has the other (but cannot see his own number), and they must try to guess the number that have on their hat. They have a timer that a bell rings every minute and the moment it rings. This is when they must say if they know the number on their hat. A third person tells them: ''the sum of the numbers is $6$ or $11$ or $19$''. At that moment it begins to run time. After a minute the bell rings and neither of them says anything. The second minute passes , the doorbell rings and neither of us says anything. Time continues to pass and when the bell rings for the tenth time Antonio says that he already knows what is his number. Just determine the number each has in his hat.

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.

Let $q<50$ be a prime number. Call a sequence of polynomials $P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)$ [i]tasty[/i] if it satisfies the following conditions: [list] [*] $P_i$ has degree $i$ for each $i$ (where we consider constant polynomials, including the $0$ polynomial, to have degree $0$) [*] The coefficients of $P_i$ are integers between $0$ and $q-1$ for each $i$. [*] For any $0\le i,j\le q^2$, the polynomial $P_i(P_j(x)) - P_j(P_i(x))$ has all its coefficients divisible by $q$. [/list] As $q$ varies over all such prime numbers, determine the total number of tasty sequences of polynomials. [i]Proposed by Vincent Huang[/i]

Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$ each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\textstyle\frac{a\sqrt b-c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$. [asy] size(200); defaultpen(linewidth(0.5)); real r=aCos(79/81); pair x=dir(270+r)*27,y=dir(270-r)*27; draw(arc(origin,30,210,330)); draw(circle(x,3)^^circle(y,3)^^circle((0,-27),3)); path arcl=arc(y,5,0,180), arcc=arc((0,-27),5,0,180), arcr=arc(x,5,0,180); pair centl=intersectionpoint(arcl,arcc), centr=intersectionpoint(arcc,arcr); draw(circle(centl,2)^^circle(centr,2)); dot(x^^y^^(0,-27)^^centl^^centr,linewidth(2)); [/asy]

The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?

Solve the following equation for $x, y \in \mathbb{N}$ \[x^3+y^3=101xy+101.\]

Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?

Knowing that $1 < y < 2$ and $x - y + 1 = 0,$ calculate the value of the expression: $$E = \sqrt{4x^2 +4y-3} + 2\sqrt{y^2 - 6x - 2y +10}.$$

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) [i]D. Tereshin[/i]

Let $p$ be a prime number and $\mathcal{A}$ be a finite set of integers, with at least $p^k$ elements. Denote by $N_{\text{even}}$ the number of subsets of $\mathcal{A}$ with even cardinality and sum of elements divisible by $p^k$. Define $N_{\text{odd}}$ similarly. Prove that $N_{\text{even}}\equiv N_{\text{odd}}\bmod{p}.$

In cube $ABCD-A_1B_1C_1D_1$, $E$ is midpoint of $BC$, $F\in AA_1$, and $A_1F:FA=1:2$. Calculate the dihedral angle between plane $B_1EF$ and plane $A_1B_1C_1D_1$.

The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$ and point $N$ is on the circle centered at $B$ such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$. [img]https://cdn.artofproblemsolving.com/attachments/3/d/1933ce259c229d49e21b9a2dcadddea2a6b404.png[/img]

Let $\{ a_n \}_{n=0}^{\infty}$, $\{ b_n \}_{n=0}^{\infty}$, and $\{ c_n \}_{n=0}^{\infty}$ be sequences of real numbers such that for all $k\geq 1$, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that $a_0=1$, $b_0=2$, and $c_0=3$. Given that $\sqrt2\approx1.4142$, compute \[ \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k). \]

Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?