On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

Let $s(n)$ denote the sum of the digits of $n$ and let $p(n)$ be the product of the digits of $n$. Find the smallest integer $k$ such that $s(k)+p(k)=49$ and $s(k+1)+p(k+1)=68$. [i]Proposed by Anthony Yang[/i]

A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? (“Opposite” means “symmetric with respect to the centre of the cube”.)

Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ . $(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$ $(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$ for the integer $k\ge 3.$

The set $X$ of integers is called [i]good[/i] If for each pair $a,b\in X$ , only one of $a+b,\mid a-b\mid$ is a member of $X$ ($a,b$ may be equal). Find the total number of sets with $2008$ as member. [i](tatari/nightmare)[/i]

Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.

Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.

[color=#000][b](i)[/b][/color] Let $a_1,a_2,...,a_n$ be n real numbers. Show that there exists some real number $\alpha$ such that $a_1+\alpha,a_2+\alpha,...,a_n+\alpha$ are all irrational. \\ [color=#000][b](ii)[/b][/color] Prove that such a satetement is not valid if all these are rquired to be rational. \\ [i]Hint (given in question) : Use Pigeon Hole Principle[/i]

Let $ABCD$ be a quadrilateral, such that $AB = BC = CD.$ There are points $X, Y$ on rays $CA, BD,$ respectively, such that $BX = CY.$ Let $P, Q, R, S$ be the midpoints of segments $BX, CY ,$ $XD, YA,$ respectively. Prove that points $P, Q, R, S$ lie on a circle.

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

An L-shape is one of the following four pieces, each consisting of three unit squares: [asy] size(300); defaultpen(linewidth(0.8)); path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1); draw(P); draw(shift((2.7,0))*rotate(90,(1,1))*P); draw(shift((5.4,0))*rotate(180,(1,1))*P); draw(shift((8.1,0))*rotate(270,(1,1))*P); [/asy] A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares. We say that a placement of L-shapes on unmarked unit squares is called $\textit{good}$ if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board. B wins if every $\textit{good}$ placement of L-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which B has a winning strategy.

Let $s(n)$ denote the sum of the digits of $n$ and let $p(n)$ be the product of the digits of $n$. Find the smallest integer $k$ such that $s(k)+p(k)=49$ and $s(k+1)+p(k+1)=68$. [i]Proposed by Anthony Yang[/i]

[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$. [b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$. [b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\] [i]Proposed by Otgonbayar Uuye. [/i]

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

$G$ be a group and $H\le G$. Then which of the following are true? $\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$ $\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$ $\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$ $\textbf{(D)}~\text{None of the above}$

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

We have an integer $A$ such that $A^2$ is a four digit number, with $5$ in the ten's place . Find all possible values of $A$.

How many pairs of integers $(a, b)$ satisfy $1 \le a < 1001^3$, $1 \le b < 1001^2$, and $1001^3 \mid a^3 + ab$?

Estimate the value of $$\sum^{2023}_{n=1} \left(1+ \frac{1}{n} \right)^n$$ to $3$ decimal places.

Positive integers $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$ satisfy $2 \leq a_i \leq 166$ and $a_i^{b_i} \cong a_{i+1}^2$ (mod 167) for each $1 \leq i \leq 7$ (where $a_8=a_1$). Compute the minimum possible value of $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$.

For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that $$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.

Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?