Find the smallest and largest values of the expression $$\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}$$ (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)

Compute the sum of all positive integers $n < 200$ such that $gcd(n, k) \ne 1$ for every $k \in\{2 \cdot 11 \cdot 19, 3 \cdot 13 \cdot 17, 5 \cdot 11 \cdot 13, 7 \cdot 17 \cdot 19\}$.

Let $a_1 < a_2 < a_3 < a_4$ be positive integers such that the following conditions hold: -$\gcd(a_i,a_j)>1$ holds for all integers $1\le i < j\le 4$. -$\gcd(a_i,a_j,a_k)=1$ holds for all integers $1\le i < j < k\le 4$. Find the smallest possible value of $a_4$. [i]Proposed by James Lin[/i]

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ .

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

Consider all cubic polynomials $f(x)$ such that $f(2018)=2018$, the graph of $f$ intersects the $y$-axis at height $2018$, the coefficients of $f$ sum to $2018$, and $f(2019)>(2018)$. We define the infinimum of a set $S$ as follows. Let $L$ be the set of lower bounds of $S$. That is, $\ell\in L$ if and only if for all $s\in S$, $\ell\leq s$. Then the infinimum of $S$ is $\max(L)$. Of all such $f(x)$, what is the infinimum of the leading coefficient (the coefficient of the $x^3$ term)?

Let $m$ and $n{}$ be positive integers of the same parity such that $n^2-1$ divides $|m^2+1-n^2|$. Is the number $|m^2+1-n^2|$ is a perfect square?

If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by $\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $

Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$ .

Suppose $x$ and $y$ are positive real numbers such that $x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$. What is the greatest possible value of $\log_2{y}$? $\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$

In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. [i]Proposed by Mahdi Etesamifard[/i]

Prove that the polynomial$$P(x)=(x-1)(x-2)(x-3)-1$$is irreducible in $\mathbb{Z}[x].$

$f$ is a polynomial of degree $n$ with integer coefficients and $f(x)=x^2+1$ for $x=1,2,\cdot ,n$. What are the possible values for $f(0)$?

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+\dfrac{1}{a_3a_4}\cdots=\dfrac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$. [i]Individual #7[/i]

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$

Secco and Ramon are drunk in the real line over the integer points $a$ and $b$, respectively. Our real line is a little bit special, though: the interval $(-\infty, 0)$ is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number $k>1$ using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point $h$ for which it wants to go. After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points $2g - h$, where $g$ is the point where each of the players were in the precedent round (that is, in the first round $g = a$ for Secco and $g = b$ for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly $k$ times bigger than the other (if both of the player end up in the point $0$, the game finishes as well). Determine, in values of $k$, the initial values $a$ and $b$ such that Secco and Ramon has a winning strategy to finish the game alive. [i]Observation: If any of the players fall in the lave, he dies and both of them lose the game[/i]

$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$ [i](S. Berlov)[/i]

Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying: (1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and (2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince. Find $|B_n^m|$ and $|B_6^3|$.

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation $$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$ for any rational numbers $ x,y. $ [i]Mihai Onucu Drîmbe[/i]

Let $a,b>1$ be odd positive integers. A board with $a$ rows and $b$ columns without fields $(2,1),(a-2,b)$ and $(a,b)$ is tiled with $2\times 2$ squares and $2\times 1$ dominoes (that can be rotated). Prove that the number of dominoes is at least $$\frac{3}{2}(a+b)-6.$$

Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and \[(10a+b)^{c/d}=e^{f/g}.\] Given that $h>i$, evaluate $10h+i$. [i]Proposed by James Lin[/i]