Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove $\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$. [i]proposed by Mohammad Ahmadi[/i]

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

[b]p1.[/b] Suppose $5$ bales of hay are weighted two at a time in all possible ways. The weights obtained are $110$, $112$, $113$, $114$, $115$, $116$, $117$, $118$, $120$, $121$. What is the difference between the heaviest and the lightest bale? [b]p2.[/b] Paul and Paula are playing a game with dice. Each have an $8$-sided die, and they roll at the same time. If the number is the same they continue rolling; otherwise the one who rolled a higher number wins. What is the probability that the game lasts at most $3$ rounds? [b]p3[/b]. Find the unique positive integer $n$ such that $\frac{n^3+5}{n^2-1}$ is an integer. [b]p4.[/b] How many numbers have $6$ digits, some four of which are $2, 0, 1, 4$ (not necessarily consecutive or in that order) and have the sum of their digits equal to $9$? [b]p5.[/b] The Duke School has $N$ students, where $N$ is at most $500$. Every year the school has three sports competitions: one in basketball, one in volleyball, and one in soccer. Students may participate in all three competitions. A basketball team has $5$ spots, a volleyball team has $6$ spots, and a soccer team has $11$ spots on the team. All students are encouraged to play, but $16$ people choose not to play basketball, $9$ choose not to play volleyball and $5$ choose not to play soccer. Miraculously, other than that all of the students who wanted to play could be divided evenly into teams of the appropriate size. How many players are there in the school? [b]p6.[/b] Let $\{a_n\}_{n\ge 1}$ be a sequence of real numbers such that $a_1 = 0$ and $a_{n+1} =\frac{a_n-\sqrt3}{\sqrt3 a_n+1}$ . Find $a_1 + a_2 +.. + a_{2014}$. [b]p7.[/b] A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon has at least two heads, the soldier is equally likely to miss or chop off two heads. The dragon dies when it has no heads left, and it overpowers the soldier if it has at least five heads. What is the probability that the soldier wins [b]p8.[/b] A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third and so on. Let $S$ be the set of integers $n$ with the property that there exists a series of moves such that after the $n$-th move the rock is back where it started. Find the number of elements in the set $S \cap \{1, 2, ..., 2014\}$. [b]p9.[/b] Find the largest integer $n$ such that the number of positive integer divisors of $n$ (including $1$ and $n$) is at least $\sqrt{n}$. [b]p10.[/b] Suppose that $x, y$ are irrational numbers such that $xy$, $x^2 + y$, $y^2 + x$ are rational numbers. Find $x + y$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S$ be the set of points $ (a,b)$ in the coordinate plane, where each of $ a$ and $ b$ may be $ \minus{} 1$, $ 0$, or $ 1$. How many distinct lines pass through at least two members of $ S$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true? $ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$

Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$ Let $E$ be the intersection of $BD$ and the internal bisector of $\angle BAD$. The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively. Prove that $A,C,X,Y$ are concyclic.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$: $$0\le y+f(x)-f^{f(y)}(x)\le1$$ that here $$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$

Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$ Note: $X^*$ is the [url = https://en.wikipedia.org/wiki/Adjugate_matrix]adjugate[/url] matrix of $X$ (not the conjugate transpose)

Points on complex plane that complex numbers $z_1,z_2$ corresponding to are $A,B$, and $|z_1|=4,4z_1^2-2z_1z_2+z_2^2=0$. $O$ is original point, then the area of $\triangle OAB$ is $\text{(A)}8\sqrt3\qquad\text{(B)}4\sqrt3\qquad\text{(C)}6\sqrt3\qquad\text{(D)}12\sqrt3$

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? $ \textbf{(A)}\ \sqrt 2\qquad \textbf{(B)}\ \sqrt 3\qquad \textbf{(C)}\ 1 \plus{} \sqrt 2\qquad \textbf{(D)}\ 1 \plus{} \sqrt 3\qquad \textbf{(E)}\ 3$

Let $A$ be an infinite subset of $\mathbb{N}$, and $n$ a fixed integer. For any prime $p$ not dividing $n$, There are infinitely many elements of $A$ not divisible by $p$. Show that for any integer $m >1, (m,n) =1$, There exist finitely many elements of $A$, such that their sum is congruent to 1 modulo $m$ and congruent to 0 modulo $n$.

In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3); draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0)); label("$A$", (.5,.5)); label("$M$", (7/6, 1/6)); label("$N$", (1/3, 4/3));[/asy] [i]Proposed by Aaron Lin[/i]

In $\vartriangle ABC, D$ and $E$ are two points on segment $BC$ such that $BD = CE$ and $\angle BAD = \angle CAE$. Prove that $\vartriangle ABC$ is isosceles

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

A positive integer is called [i]squarefree[/i] if its only perfect square factor is $1$. Call a set of positive integers [i]squarefreeful[/i] if each product of two of its elements is squarefree, and [i]squarefreefullest[/i] if no positive integer less than the maximum element of the set can be added while preserving the set's squarefreefulness. What is the minimum number of elements in a squarefreefullest set containing $31$?

A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$

Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;

Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$. Then the value of $G$ is?

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$. [i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]

Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$ for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).

Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.

The sequence $\{a_n\}_{n\geq 1}$ is defined by $a_{n+2}=7a_{n+1}-a_n$ for positive integers $n$ with initial values $a_1=1$ and $a_2=8$. Another sequence, $\{b_n\}$, is defined by the rule $b_{n+2}=3b_{n+1}-b_n$ for positive integers $n$ together with the values $b_1=1$ and $b_2=2$. Find $\gcd(a_{5000},b_{501})$.

Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$.

Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$ Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$