Prove that the equation \[\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] has infinitly many natural solutions

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions: i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$ ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$ Determine the minimum value of $m$.

There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.

Fill in each cell of the grid with a positive digit so that the following conditions hold: 1. each row and column contains ve distinct digits; 2. for any cage containing multiple cells of a row, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole row, and 3. for any cage containing multiple cells of a column, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole column. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [asy] unitsize(48); int[][] a = { {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}; for (int i = 0; i < 5; ++i) { for (int j = 0; j < 5; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(24pt)); } } real ep=0.1; real s=3; pen lw=linewidth(.12mm); real x=0.9; real y=1.2; draw((0+s*ep,0-ep)--(2-ep,0-ep)--(2-ep,-1+ep)--(0+ep,-1+ep)--(0+ep,0-s*ep),dashed+lw); draw((2+s*ep,0-ep)--(4-ep,0-ep)--(4-ep,-1+ep)--(2+ep,-1+ep)--(2+ep,0-s*ep),dashed+lw); draw((1+s*ep,-1-ep)--(3-ep,-1-ep)--(3-ep,-2+ep)--(1+ep,-2+ep)--(1+ep,-1-s*ep),dashed+lw); draw((2+s*ep,-3-ep)--(4-ep,-3-ep)--(4-ep,-4+ep)--(2+ep,-4+ep)--(2+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-4-ep)--(3-ep,-4-ep)--(3-ep,-5+ep)--(1+ep,-5+ep)--(1+ep,-4-s*ep),dashed+lw); draw((3+s*ep,-4-ep)--(5-ep,-4-ep)--(5-ep,-5+ep)--(3+ep,-5+ep)--(3+ep,-4-s*ep),dashed+lw); label(scale(x)*"5", (0+ep,0-y*ep)); label(scale(x)*"7", (2+ep,0-y*ep)); label(scale(x)*"10", (1+ep,-1-y*ep)); label(scale(x)*"5", (2+ep,-3-y*ep)); label(scale(x)*"2", (1+ep, -4-y*ep)); label(scale(x)*"13", (3+ep, -4-y*ep)); draw((4+s*ep,0-ep)--(5-ep,0-ep)--(5-ep,-2+ep)--(4+ep,-2+ep)--(4+ep,0-s*ep),dashed+lw); draw((0+s*ep,-1-ep)--(1-ep,-1-ep)--(1-ep,-3+ep)--(0+ep,-3+ep)--(0+ep,-1-s*ep),dashed+lw); draw((3+s*ep,-1-ep)--(4-ep,-1-ep)--(4-ep,-3+ep)--(3+ep,-3+ep)--(3+ep,-1-s*ep),dashed+lw); draw((0+s*ep,-3-ep)--(1-ep,-3-ep)--(1-ep,-5+ep)--(0+ep,-5+ep)--(0+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-2-ep)--(2-ep,-2-ep)--(2-ep,-4+ep)--(1+ep,-4+ep)--(1+ep,-2-s*ep),dashed+lw); draw((4+s*ep,-2-ep)--(5-ep,-2-ep)--(5-ep,-4+ep)--(4+ep,-4+ep)--(4+ep,-2-s*ep),dashed+lw); label(scale(x)*"10", (4+ep,0-y*ep)); label(scale(x)*"3", (0+ep,-1-y*ep)); label(scale(x)*"8", (3+ep,-1-y*ep)); label(scale(x)*"16", (1+ep,-2-y*ep)); label(scale(x)*"6", (4+ep,-2-y*ep)); label(scale(x)*"11", (0+ep,-3-y*ep)); [/asy]

Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$

ff nonnegative real numbers$ x_1,x_2,...,x_n$ satisfy $x_1 +...+x_n\le \frac12$, prove that $$(1-x_1)(1-x_2)...(1-x_n) \ge \frac12$$

The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that $AB$ goes to $DA$ $DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$ then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?

A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule \[a_{n+1}=a_n+b_n\ (n=1,2,\ldots)\] where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.

Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$. [i]V. Kamianetski[/i]

A Tetris Figure is every figure in the plane which consists of $4$ unit squares connected by their sides (and don’t overlap). Two Tetris Figures are the same if one can be rotated in the plane to become the other. a) Prove that there exist exactly $7$ different Tetris Figures. b) Is it possible to fill a $4 \times 7$ rectangle by using once each of the $7$ different Tetris Figures?

For $p$, $q$, $l$ strictly positive real numbers, consider the following problem: for $y \geq 0$ fixed, determine the values $x \geq 0$ such that $x^p - lx^q \leq y$. Denote by $S(y)$ the set of solutions of this problem. Prove that if one has $p < q$, $\epsilon \in (0, l^\frac{1}{p-q})$, $0 \leq x \leq \epsilon$ and $x \in S(y)$, then $$x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}$$

Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]

What value of $x$ satisfies \[\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?\] $ \textbf{(A) }25\qquad \textbf{(B) }32\qquad \textbf{(C) }36\qquad \textbf{(D) }42\qquad \textbf{(E) }48\qquad $

[u]Round 1[/u] [b]p1.[/b] Alice and Bob played $25$ games of rock-paper-scissors. Alice played rock $12$ times, scissors $6$ times, and paper $7$ times. Bob played rock $13$ times, scissors $9$ times, and paper $3$ times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.) [b]p2.[/b] On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were $30$ eruptions. How many volcanoes could erupt this year? [b]p3.[/b] A tangle is a sequence of digits constructed by picking a number $N\ge 0$ and writing the integers from $0$ to $N$ in some order, with no spaces. For example, $010123459876$ is a tangle with $N = 10$. A palindromic sequence reads the same forward or backward, such as $878$ or $6226$. The shortest palindromic tangle is $0$. How long is the second-shortest palindromic tangle? [b]p4.[/b] Balls numbered $1$ to $N$ have been randomly arranged in a long input tube that feeds into the upper left square of an $8 \times 8$ board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from $1$ to $N$ in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number $N$ for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. [img]https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png[/img] [b]p5.[/b] A $2018 \times 2018$ board is covered by non-overlapping $2 \times 1$ dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? [img]https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png[/img] [u]Round 2[/u] [b]p6.[/b] Seventeen teams participated in a soccer tournament where a win is worth $1$ point, a tie is worth $0$ points, and a loss is worth $-1$ point. Each team played each other team exactly once. At least $\frac34$ of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament. [b]p7.[/b] The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are $2018$ influential secret societies. Show that it is possible to form a council of at most $11$ people such that each influential secret society has at least one member on the council. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b) \mid af(a) - bf(b)$, for all $a, b \in \mathbb{N}$. (Here, $\mathbb{N}$ is a set of positive integers.) [i]Proposed by Vukašin Pantelić[/i]

Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

$a,b,c$ - different natural numbers. Can we build quadratic polynomial $P(x)=kx^2+lx+m$, with $k,l,m$ are integer, $k>0$ that for some integer points it get values $a^3,b^3,c^3$ ?

Billy is standing at $(1,0)$ in the coordinate plane as he watches his Aunt Sydney go for her morning jog starting at the origin. If Aunt Sydney runs into the First Quadrant at a constant speed of $1$ meter per second along the graph of $x=\frac25y^2$, find the rate, in radians per second, at which Billy’s head is turning clockwise when Aunt Sydney passes through $x=1$.

Define a positive integer $n$ to be [i]almost-cubic [/i] if it becomes a perfect cube upon concatenating the digit $5$. For example, $12$ is almost-cubic because $125 = 5^3$. Find the remainder when the sum of all almost-cubic $n < 10^8$ is divided by $1000$.

[b]9.[/b] Show that to any elliptic paraboloid $\varphi_1$ there may be found an elliptic paraboloid $\varphi_2$ (other than $\varphi_1$) and an affinity $\phi$ which maps $\varphi_1$ onto $\varphi_2$ and has the following property: If $P$ is any point of $\varphi_1$ such that $\phi(P) \neq P$, then the straight line connecting $P$ and $\phi(P)$ is a common tangent of the two paraboloids. [b](G. 18)[/b]

Let $n$ be a natural number and the polynomial, $P(x)=x^n+n$. $(a)$ Is it possible that for some odd number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$. $(b)$ Is it possible that for some even number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$. Reason your answers.

Positive integers $1,2,...,n$ are written on а blackboard ($n >2$ ). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least $n$ for which it is possible.

$H$ is the orthocenter of acude triangle $ABC$.Let $\omega$ be the circumcircle of $BHC$ with center $O'$.$\Omega$ is the nine-point circle of $ABC$.$X$ is an arbitrary point on arc $BHC$ of $\omega$ and $AX$ intersects $\Omega$ at $Y$.$P$ is a point on $\Omega$ such that $PX=PY$.Prove that $O'PX=90$.

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.