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.

Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent: (a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$. (b) $(A,+,\cdot)$ is a field.

[list=a] [*] Find an example of three positive integers $a,b,c$ satisfying $31a+30b+28c=365$. [*] Prove that any triplet $a,b,c$ satisfying the above condition, also satisfies $a+b+c=12$. [/list]

There are $2024$ mathematicians sitting in a row next to the river Tisza. Each of them is working on exactly one research topic, and if two mathematicians are working on the same topic, everyone sitting between them is also working on it. Marvin is trying to figure out for each pair of mathematicians whether they are working on the same topic. He is allowed to ask each mathematician the following question: “How many of these 2024 mathematicians are working on your topic?” He asks the questions one by one, so he knows all previous answers before he asks the next one. Determine the smallest positive integer $k$ such that Marvin can always accomplish his goal with at most $k$ questions.

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt 6$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt 3$. What is $|AC|$ ? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 12 $

Let $ n \geq 3$ be an odd integer. We denote by $ [\minus{}n,n]$ the set of all integers greater or equal than $ \minus{}n$ and less or equal than $ n$. Player $ A$ chooses an arbitrary positive integer $ k$, then player $ B$ picks a subset of $ k$ (distinct) elements from $ [\minus{}n,n]$. Let this subset be $ S$. If all numbers in $ [\minus{}n,n]$ can be written as the sum of exactly $ n$ distinct elements of $ S$, then player $ A$ wins the game. If not, $ B$ wins. Find the least value of $ k$ such that player $ A$ can always win the game.

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.

Consider the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \in \{-1, 0, 1\}$ and $(x,y) \neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints? [i]Proposed by Evan Chen[/i]

Find all functions $f$ from real numbers to real numbers such that $$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$$ holds for all real numbers $x$ and $y$.

Prove that if $ n $ is a natural number greater than $ 2 $, then $$\sqrt[n + 1]{n+1} < \sqrt[n]{n}.$$

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.

An acute angled triangle $\mathcal{T}$ is inscribed in circle $\Omega$.Denote by $\Gamma$ the nine-point circle of $\mathcal{T}$.A circle $\omega$ passes through two of the vertices of $\mathcal{T}$, and centre of $\Omega$.Prove that the common external tangents of $\Gamma$ and $\omega$ meet on the external bisector of the angle at third vertex of $\mathcal{T}$.

Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 10 $