Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

An acute triangle $ABC$ has $AB$ as one of its longest sides. The incircle of $ABC$ has center $I$ and radius $r$. Line $CI$ meets the circumcircle of $ABC$ at $D$. Let $E$ be a point on the minor arc $BC$ of the circumcircle of $ABC$ with $\angle ABE > \angle BAD$ and $E\notin \{B,C\}$. Line $AB$ meets $DE$ at $F$ and line $AD$ meets $BE$ at $G$. Let $P$ be a point inside triangle $AGE$ with $\angle APE=\angle AFE$ and $P\neq F$. Let $X$ be a point on side $AE$ with $XP\parallel EG$ and let $S$ be a point on side $EG$ with $PS\parallel AE$. Suppose $XS$ and $GP$ meet on the circumcircle of $AGE$. Determine the possible positions of $E$ as well as the minimum value of $\frac{BE}{r}$.

Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$

Amélia and Beatriz play battleship on a $2n\times2n$ board, using very peculiar rules. Amélia begins by choosing $n$ lines and $n$ columns of the board, placing her $n^2$ submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?

Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest . Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$. In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.

Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel. [i]Proposed by Samuel Wang[/i] [hide=Solution][i]Solution.[/i] $\boxed{1000001}$ Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 = \boxed{1000001}$.[/hide]

The bivariate functions $f_0, f_1, f_2, f_3, \dots$ are sequentially defined by the relations $f_0(x,y) = 0$ and $f_{n+1}(x,y) = \bigl|x+|y+f_n(x,y)|\bigr|$ for all integers $n \geq 0$. For independently and randomly selected values $x_0, y_0 \in [-2, 2]$, let $p_n$ be the probability that $f_n(x_0, y_0) < 1$. Let $a,b,c,$ and $d$ be positive integers such that the limit of the sequence $p_1,p_3,p_5,p_7,\dots$ is $\frac{\pi^2+a}{b}$ and the limit of the sequence $p_0,p_2,p_4,p_6,p_8, \dots$ is $\frac{\pi^2+c}{d}$. Compute $1000a+100b+10c+d$. [i]Proposed by Sean Li[/i]

Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$ For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$ where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$. Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd [i]M. Zorka[/i]

Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the permutation, the number of numbers less than $k$ that follow $k$ is even. For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab$.

Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that \[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$. Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.

Let be three real positive numbers $ \alpha ,\beta ,\gamma $ and let $ M,N $ be points on the sides $ AB,BC, $ respectively, of a triangle $ ABC, $ such that $ \frac{MA}{MB} =\frac{\alpha }{\beta } $ and $ \frac{NB}{NC} =\frac{\beta }{\gamma } . $ Also, let $ P $ be the intersection of $ CM $ with $ AN. $ Show that: $$ \frac{1}{\alpha }\overrightarrow{PA} +\frac{1}{\beta }\overrightarrow{PB} +\frac{1}{\gamma }\overrightarrow{PC} =0 $$

Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.

We call Golomb ruler a ruler of length $l$, bearing $k+1\geq 2$ marks $0<a_1<\ldots <a_{k-1}<l$, such that the lengths that can be measured using marks on the ruler are consecutive integers starting with $1$, and each such length be measurable between just two of the gradations of the ruler. Find all Golomb rulers.

There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!) (a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$. (b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$. [i]Proposed by Ray Li[/i]

Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.

Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.

Real numbers $a,b,c,d$ such that $|a|>1$ , $|b|>1$ , $|c|>1$ , $|d|>1$ and $ab(c+d)+dc(a+b)+a+b+c+d=0$ then prove that $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}+\frac{1}{d-1} >0$

For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.

Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.