Line $l:2x+y=10$, line $l'$ passes $(-10,0)$, and $l'\perp l$, then the coordinate of the intersection of $l$ and $l'$ is________.

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

Construct the $ \triangle ABC $, with $ AC <BC $, if the circumcircle is known, and the points $ D, E, F $ in it, where they intersect, respectively, the altitude, the median and the angle bisector that they start from the vertex $ C $.

Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

Find $a$ and $b$ for which the largest and smallest is values of the function $y=\frac{x^2+ax+b}{x^2-x+1}$ are equal to the $2$ and $-3$ respectively.

Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.

The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

For every $0 < \alpha < 1$, let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$. Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$, that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$. Determine the area of $R$.

Determine all positive integers $n\ge 3$ such that the inequality \[a_1a_2+a_2a_3+\ldots a_{n-1}a_n\le 0\] holds for all real numbers $a_1,a_2,\ldots , a_n$ which satisfy $a_1+a_2+\ldots +a_n=0$.

Determine all pairs ($m,n$) of natural numbers for which $4^m + 5^n$ is a perfect square.

There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$.

Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.

Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's comic book collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25$

Triangle $ABC$ in which $AB <BC$, is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. The line $\ell$ parallel to $AC$, touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. It is known that $PQ = 2BI$. Prove that $AP + 2PB = CP$.

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$. [i]Aleksandar Ivanov[/i]

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

$A$ is a non-empty subset of positive integers. Let $$f(A)=\{abc-b-c+2\vert a,b,c\in A\}$$ Determine all integers $n$ greater than $1$ so that we can divide the set of positive integers into $A_1, A_2, \dots, A_n$ ($A_i\neq \emptyset (i=1, 2, \dots , n)$, $\forall 1\le i < j \le n, A_i\cap A_j = \emptyset$ and $\bigcup_{i=1}^{n} A_i=\mathbb{N}^*$) satisfy that $\forall 1\le i\le n, f(A_i) \subseteq A_i$.

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

Let $\omega$ be a root of unity and $f$ be a polynomial with integer coefficients. Show that if $|f(\omega)|=1$, then $f(\omega)$ is also a root of unity.

How many ways are there to permute the letters $\{S,C,R, A,M,B,L,E\}$ without the permutation containing the substring $L AME$?

Given an integer $n > 1$, let $1 = a_1 < a_2 < \cdots < a_t = n - 1$ be all positive integers less than $n$ that are coprime to $n$. Find all $n$ such that there is no $i \in \{1, 2, \ldots , t - 1\}$ satisfying $3 | a_i + a_{i+1}$.

Melanie has $4\frac{2}{5}$ cups of flour. The recipe for one batch of cookies calls for $1\frac{1}{2}$ cups of flour. Melanie plans to make $2\frac{1}{2}$ batches of cookies. When she is done, she will have $\frac{m}{n}$ cups of flour remaining, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $a$ be a real number. Find all real-valued functions $f$ such that $$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$ when constants of integration are suitably chosen.