Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\] [i]Proposed by Japan[/i]

You own two cats, Chocolate and Tea. Chocolate and Tea sleep for $C$ and $T$ hours a day respectively, where $C$ and $T$ are chosen independently and uniformly at random from the interval $[5,10]$. In a given day, what is the probability that Chocolate and Tea will together sleep for a total of at least $14$ hours?

Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$. The tangent line to from $A$ to $\omega$ intersects $BC$ at $K$. The tangent line to from $B$ to $\omega$ intersects $AC$ at $L$. Let $M,N$ be the midpoints of $AK,BL$ respectively. The line $MN$ is named by $\alpha$. The feet of perpendicular from $A,B,C$ to the edges of $\triangle ABC$ are named by $D,E,F$ respectively. The perpendicular bisectors of $EF,DF,DE$ intersect $\alpha$ at $X,Y,Z$ respectively. Let $AD,BE,CF$ intersect $\omega$ again at $D',E',F'$ respectively. If $H$ is the orthocenter of $ABC$, prove that the lines $XD',YE',ZF',OH$ are concurrent.

Suppose that $P(x)$ is a polynomial with real coefficients, such that for some positive real numbers $c$ and $d$, and for all natural numbers $n$, we have $c|n|^3\leq |P(n)|\leq d|n|^3$. Prove that $P(x)$ has a real zero.

Cut the $10$ cm $\times 25$ cm rectangle into two pieces with one straight cut so that they can fit inside the $22.1 $ cm circle without crossing.

Let \( p \geq 7 \) be a prime number and \( m \in \mathbb{N} \). Prove that \[\left| p^m - (p - 2)! \right| > p^2.\] [i]Proposed by Miloš Milićev[/i]

On each edge of a regular tetrahedron, five points that separate the edge into six equal segments are marked. There are twenty planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these twenty planes, how many new tetrahedrons are produced?

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by $n$ an integer that is shown on the calculator. A person types an integer, $m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the first $99$ positive integers, and if $m\%$ of the number $n$ is again a positive integer, then the calculator displays $m\%$ of $n$. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from $\{1, 2, . . . , 2019\}$ guarantee the winning strategy for the statistician, who plays second? For example, if the calculator displays $1200$, the economist can type $50$, giving the number $600$ on the calculator, then the statistician can type $25$ giving the number $150$. Now, for instance, the economist cannot type $75$ as $75\%$ of $150$ is not a positive integer, but can choose $40$ and the game continues until one of them cannot type an allowed number [i]Proposed by Serbia [/i]

Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$ [i](8 points)[/i]

Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2,\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those $n - 2$ products. Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incenter of $ABC$. Draw a line perpendicular to $AI$ at $I$. Let it intersect the line $CB$ at $D$. Prove that $CI$ is perpendicular to $AD$ and prove that $ID=\sqrt{b(b-a)}$ where $BC=a$ and $CA=b$.

$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$ $\textbf{a)}$ Prove that the given sequence is an arithmetic progression. $\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.

Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Prove that there exists and infinity of multiples of $1997$ that have $1998$ as first four digits and last four digits.

A sequence of real numbers $\{a_n\}_n$ is called a [i]bs[/i] sequence if $a_n = |a_{n+1} - a_{n+2}|$, for all $n\geq 0$. Prove that a bs sequence is bounded if and only if the function $f$ given by $f(n,k)=a_na_k(a_n-a_k)$, for all $n,k\geq 0$ is the null function. [i]Mihai Baluna - ISL 2004[/i]

A set of 1990 persons is divided into non-intersecting subsets in such a way that 1. No one in a subset knows all the others in the subset, 2. Among any three persons in a subset, there are always at least two who do not know each other, and 3. For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them. (a) Prove that within each subset, every person has the same number of acquaintances. (b) Determine the maximum possible number of subsets. Note: It is understood that if a person $A$ knows person $B$, then person $B$ will know person $A$; an acquaintance is someone who is known. Every person is assumed to know one's self.

Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.

Consider the equation \[x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0\] where $N$ is a given positive integer. a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$), b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.

A ream of paper containing $ 500$ sheets is $ 5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $ 7.5$ cm high? \[ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 550 \qquad \textbf{(C)}\ 667 \qquad \textbf{(D)}\ 750 \qquad \textbf{(E)}\ 1250 \]

Suppose that $a$ and $b$ are positive integers such that $\gcd(a^3 - b^3,(a-b)^3)$ is not divisible by any perfect square except $1.$ Given that $1 \le a-b \le 50,$ compute the number of possible values of $a-b$ across all such $a,b.$

Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find $ n$.

We have $2015$ marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take $2$ different colored marbles, then replace it with $2$ marbles with the third color. For example, we take one blue marble and one green marble, and we fill with $2$ red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.

Given $k\ge 2$, for which polynomials $P\in \mathbb{Z}[X]$ does there exist a function $h:\mathbb{N}\rightarrow\mathbb{N}$ with $h^{(k)}(n)=P(n)$?