$ a_1, a_2, \cdots, a_n$ is a sequence of real numbers, and $ b_1, b_2, \cdots, b_n$ are real numbers that satisfy the condition $ 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0$. Prove that there exists a natural number $ k \le n$ that satisifes $ |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|$

Evaluate $$\lim_{n\to\infty}\left(\frac{1+\frac13+\ldots+\frac1{2n+1}}{\ln\sqrt n}\right)^{\ln\sqrt n}$$ [i]Proposed by Ángel Plaza[/i]

For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

A number of $17$ workers stand in a row. Every contiguous group of at least $2$ workers is a $\textit{brigade}$. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by $4$. Prove that the number of such ways to assign the leaders is divisible by $17$. [i]Mikhail Antipov, Russia[/i]

In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$

A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle.

If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$ .

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$

In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.

The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$, $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$.

Let $ABCD$ be a cyclic quadrilateral with center $O$ with $AB > CD$ and $BC > AD$. Let $M$ and $N$ be the midpoint of sides $AD$ and $BC$, respectively, and let $X$ and $Y$ be on $AB$ and $CD$, respectively, such that $AX \cdot CY = BX \cdot DY = 20000$, and $AX \le CY$. Let lines $AD$ and $BC$ hit at $P$, and let lines $AB$ and $CD$ hit at $Q$. The circumcircles of $\triangle MNP$ and $\triangle XYQ$ hit at a point $R$ that is on the opposite side of $CD$ as $O$. Let $R_1$ be the midpoint of $PQ$ and $B$, $D$, and $R$ be collinear. Let $O_1$ be the circumcenter of $\triangle BPQ$. Let the lines $BO_1$ and $DR_1$ intersect at a point $I$. If $BP \cdot BQ = 823875$, $AB=429$, and $BC=495$, then $IR=\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a,c) = 1$. Find the value of $a+b+c$. [i]Proposed by Kevin Zhao[/i]

Let $ABC$ be an acute triangle with circumcircle $\omega$ and orthocenter $H$. Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3$, $HX=2$, $HY=6$. The area of $\triangle ABC$ can be written as $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares. $(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? $(b)$ When it is possible, find the minimum number of trominoes needed.

Find all distinct prime numbers $p,q,r,s$ such that $1-\frac{1}{p} - \frac{1}{q} -\frac{1}{r} - \frac{1}{s} =\frac{1}{pqrs}$

Define a sequence called the $2020$-nacci sequence. It is defined as follows: If $n \le 2020$ then $S_n=1,$ if $n>2020$ then $S_n=\sum_{i=n-2020}^{n-1} S_i.$ Find the last two digits of $S_{4040}.$

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

Let $M$ be a set consisting of $n$ points in the plane, satisfying: i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon; ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon. Find the minimum value of $n$. [i]Leng Gangsong[/i]

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by \[ f(x)=\det(A+xB)\] has a multiple root, then $ \det(A+B)=\det B$.

A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.

Suppose $a$ and $b$ are positive integers for which $8a^ab^b = 27a^bb^a$. Find $a^2 + b^2$.

Let $\triangle BC$ be a triangle with side lengths $AB = 9, BC = 10, CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC, E = BO \cap CA, F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

The sides of triangle $BAC$ are in the ratio $2: 3: 4$. $BD$ is the angle-bisector drawn to the shortest side $AC$, dividing it into segments $AD$ and $CD$. If the length of $AC$ is $10$, then the length of the longer segment of $AC$ is: $\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12$

Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying : $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$