Area of convex quadrilateral $ABCD$ is $1$. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than $\frac{1}{4}$.

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

Prove that all acute-angled triangles with the equal altitudes $h_c$ and the equal angles $\gamma$ have orthic triangles with same perimeters.

Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.

Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?

Find the number of two-digit positive integers whose digits total $7$. $\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

Let $AB$ be a chord of the semicircle $O$ (not the diameter). $M$ is the midpoint of $AB$, and $D$ is a point lies on line $OM$ ($D$ is outside semicircle $O$). Line $l$ passes through $D$ and is parallel to $AB$. $P, Q$ are two points lie on $l$ and $PO$ meets semicircle $O$ at $C$. If $\angle PCD=\angle DMC$, and $M$ is the orthocentre of $\triangle OPQ$. Prove that the intersection of $AQ$ and $PB$ lies on semicircle $O$.

Let $W,A,B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent. [list] [*] For all $x, y, z\ge 0$ satisfying $x+y\le z+W, x+z\le y+W, y+z\le x+W$ we have $Axyz+B\ge x^2+y^2+z^2$. [*] $B\ge W^2$ and $AW^3+B\ge 3W^2$. [/list] [i]Proposed by Ákos Somogyi, London[/i]

A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules. [list] [*]The binary string $1$ is in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]No other binary strings are in $\mathcal{B}$.[/*] [/list] For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$. [list=a] [*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*] [*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*] [/list] [i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]

Prove the identity \[ 1 \plus{} \frac{1}{2} \minus{} \frac{2}{3} \plus{} \frac{1}{4} \plus{} \frac{1}{5} \minus{} \frac{2}{6} \plus{} \ldots \plus{} \frac{1}{478} \plus{} \frac{1}{479} \minus{} \frac{2}{480} \equal{} 2 \cdot \sum^{159}_{k\equal{}0} \frac{641}{(161\plus{}k) \cdot (480\minus{}k)}.\]

Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.

A circle $\omega$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $XP$ and the circle $XOP$ meet $\omega$ for a second time at points $X_1$, $X_2$ respectively. Prove that all lines $X_1X_2$ are parallel.

Find the number of pairs of integers $(a,b)$ with $0 \le a,b \le 2019$ where $ax \equiv b \pmod{2020}$ has exactly $2$ integer solutions $0 \le x \le 2019$. [i]Proposed by Richard Chen[/i]

Let $(a_n)_{n\ge 1}$ be an increasing sequence of positive integers. Assume that there is a constant $M>0$ satisfying$$0<a_{n+1}-a_n<M.a_n^{5/8},\forall n\ge 1.$$ Prove that: there exists a real number $A$ such that for each $k\in \mathbb{Z}^+,[A^{3^k}]$ is an element of $(a_n)_{n\ge 1}.$

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$. (Karl Czakler)

Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$ is an integer and find its value