Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.

The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.

Find all pair $(m,n)$ of integer ($m >1,n \geq 3$) with the following property:If an $n$-gon can be partitioned into $m$ isoceles triangles,then the $n$-gon has two congruent sides.

Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.

Suppose that $a_1,a_2,\dots a_n$ are real $(n>1)$ and $$A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.$$ Prove that $A<2a_ia_j$ for $1\leq i<j\leq n.$

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$

Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$.

Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$. (i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$. (ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted: 1. Each person moves to the next seat clockwise. 2. Only the ones in seats 1 and $k$ exchange their seats. Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$?

Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.

Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this. Note that boy A doesn’t have to return to the starting point to leave gotten flags.

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the ”order” $n$, when in this point $n$ line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination $\{2,2,2,2,2,2\}$ and a star with six rays by $\{6,1,1,1,1,1,1\}$. (a) Sketch a figure' belonging to the combination $\{4,3,3,3,3\}$. (b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to $6$. (c) Prove that every such combination contains an even number of odd numbers.

Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled? $ \text{(A)}\ 20\qquad\text{(B)}\ 24\qquad\text{(C)}\ 32\qquad\text{(D)}\ 33\qquad\text{(E)}\ 40 $

Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$ Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$

[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$? [b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight? [b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together? [b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible. [b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer? [b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time? [b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$. [b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)? [b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. [b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$? [b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

We define a complex number $z=9+10i$ please find the maximum of a positive integer $n$ which satisfies $|z^n|\leq2023$

Find the highest power of 3 dividing $\dbinom{666}{333}$.

Three students start walking with constant speeds at the same time, each along a straight line in the plane. Prove that if the students are not on the same line at the beginning, then they will be on the same line at most twice during their journey.

The integers $1, 2, 3, 4, 5$ and $6$ are written on a board. You can perform the following kind of move: select two of the numbers, say $a$ and $b$, such that $4a - 2b$ is nonnegative; erase $a$ and $b$, then write down $4a - 2b$ on the board (hence replacing two of the numbers by just one). Continue performing such moves until only one number remains on the board. What is the smallest possible positive value of this last remaining number?

Fix an integer $n \ge 2$. A fairy chess piece [i]leopard [/i] may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3n \times 3n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made. Dmitry Khramtsov, Russia