Find all real solutions $x$ of the equation $$\lfloor x^2-2 \rfloor +2 \lfloor x \rfloor = \lfloor x \rfloor ^2. $$ .

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

Batman, who has a mass of $\text{M = 100 kg}$, climbs to the roof of a $\text{30 m}$ building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of $\text{m = 75 kg}$, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof? (A) $\text{60 J}$ (B) $\text{7} \times \text{10}^3 \text{J}$ (C) $\text{5} \times \text{10}^4 \text{J}$ (D) $\text{600 J}$ (E) $\text{3} \times \text{10}^4 \text{J}$

Determine all real numbers $x$ and $y$ such that $x^2 + x = y^3 - y$, $y^2 + y = x^3 - x$

Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, $1, 2, 3, \ldots$ All of the lockers are initially closed. Initially, he has some set $S = \{1, 2, 3, \ldots\}$. Every step, for each element $s$ of $S$, Grant goes through the hallway and opens each locker divisible by $s$ that is closed, and closes each locker divisible by $s$ that is open. Once he does this for all $s$, he then replaces $S$ with the set of labels of the currently open lockers, and then closes every door again. After $2022$ steps, $S$ has $n$ integers that divide ${10}^{2022}$. Find $n$. [i]Proposed by Oliver Hayman[/i]

Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers. Calculate $\det(A)$.

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system: \[a_1^2 + a_1 - 1 = a_2\] \[ a_2^2 + a_2 - 1 = a_3\] \[\hspace*{3.3em} \vdots \] \[a_{n}^2 + a_n - 1 = a_1\]

Some positive integer has an even number of divisors. Anya wants to split these divisors into pairs so that the products of the numbers in each pair have the same number of divisors. Prove that she can do this in exactly one way. [i]Proposed by Oleksii Masalitin[/i]

Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $ Prove that $ 0=\det\left( A+\text{adj} A \right) . $ [i]Neculai Moraru[/i]

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

For each positive integer $k$, define $a_k$ as the number obtained from adding $k$ zeroes and a $1$ to the right of $2024$, all written in base $10$. Determine wether there's a $k$ such that $a_k$ has at least $2024^{2024}$ distinct prime divisors.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ? $ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$

$100$ Queens are placed on a $100 \times 100$ chessboard so that no two attack each other. Prove that each of four $50 \times 50$ corners of the board contains at least one Queen.

$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.

What is the value in simplest form of the following expression? \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\] $\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(f(x-y)-yf(x))=xf(y).$$

Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$? $ \textbf{(A)}\ \sqrt2 \qquad\textbf{(B)}\ \frac{2\sqrt5}{3} \qquad\textbf{(C)}\ \frac{3\sqrt5}{4} \qquad\textbf{(D)}\ \sqrt3 \qquad\textbf{(E)}\ \frac{2\sqrt7}{3} $

Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Powell Zhang[/i]

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

Rectangle $ABCD$ measures $70$ by $40$. Eighteen points (including $A$ and $C$) are marked on the diagonal $AC$ dividing the diagonal into $17$ congruent pieces. Twenty-two points (including A and B) are marked on the side $AB$ dividing the side into $21$ congruent pieces. Seventeen non-overlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left $17$ of the $21$ congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these $17$ triangles. [asy] size(200); defaultpen(linewidth(0.8)); pair A=origin,B=(21,0),C=(21,12),D=(0,12); path P=origin; draw(A--B--C--D--cycle--C); for (int r = 1; r <= 17;++r) { P=P--(21*r/17,12*r/17)--(r,0); } P=P--cycle; filldraw(P,gray(0.7)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); [/asy]

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]