For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

The problems are really difficult to find online, so here are the problems. P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form. a) Determine the smallest possible value of $k$. b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$. c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4. I added this because my translation is a bit weird. [hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana. a) Tentukan bilangan $k$ terkecil yang mungkin. b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$. c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide] P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs. [hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description: For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0). Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve. Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide] Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$. a) Draw the graph of the function $g \circ f$. b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$. P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$. P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$. P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.

Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation.

How many sets of positive integers $(a,b,c)$ satisfy $a>b>c>0$ and $a+b+c=103$?

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.

The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.

The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$ over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$

Show that one can find $50$ distinct positive integers such that the sum of each number and its digits is the same.

A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?

A boy write three times the natural number $n$ in a blackboard. He then performed an operation of the following type several times: He erased one of the numbers and wrote in its place the sum of the two others minus $1$. After several moves, one of the three numbers in the blackboard is $900$. Find all the posible values of $n$.

Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number $$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$

Two circles $\alpha{}$ and $\beta{}$ with centers $A{}$ and $B{}$ respectively intersect at points $C{}$ and $D{}$. The segment $AB{}$ intersects $\alpha{}$ and $\beta{}$ at points $K{}$ and $L{}$ respectively. The ray $DK$ intersects the circle $\beta{}$ for the second time at the point $N{}$, and the ray $DL$ intersects the circle $\alpha{}$ for the second time at the point $M{}$. Prove that the intersection point of the diagonals of the quadrangle $KLMN$ coincides with the incenter of the triangle $ABC$. [i]Konstantin Knop[/i]

Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.

Let $\omega$ be a circle with diameter $\overline{AB},$ center $O,$ and cyclic quadrilateral $ABCD$ inscribed in it, with $C$ and $D$ on the same side of $\overline{AB}.$ Let $AB=20, BC=13, AD=7.$ Let $\overleftrightarrow{BC}$ and $\overleftrightarrow{AD}$ intersect at $E.$ Let the $E$-excircle of $ECD$ have its center at $L.$ Find $OL.$

The incenter $O$ of an isosceles triangle $ABC$ with $AB=AC$ meets $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. Let $N$ be the intersection of lines $OL$ and $KM$ and let $Q$ be the intersection of lines $BN$ and $CA$. Let $P$ be the foot of the perpendicular from $A$ to $BQ$. If we assume that $BP=AP+2PQ$, what are the possible values of $\frac{AB}{BC}$?

A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction? (B, Frenkin)

The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions: a) No pair of islands is joined by more than one bridge. b) Using the bridges we can reach every island from Panacenter. c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing. Determine the number of ways we can build the bridges.

Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset.

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1 + 2 + \cdots + k = (k + 1) + (k + 2)+\cdots + N.$