Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$. a) Prove that $\frac{s}{r}$ is a natural number. b) Show that both $r$ and $s$ are natural numbers. Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.

Points $ A,B,C,D,E$ and $ F$ lie, in that order, on $ \overline{AF}$, dividing it into five segments, each of length 1. Point $ G$ is not on line $ AF$. Point $ H$ lies on $ \overline{GD}$, and point $ J$ lies on $ \overline{GF}$. The line segments $ \overline{HC}, \overline{JE},$ and $ \overline{AG}$ are parallel. Find $ HC/JE$. $ \text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2$

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Let $ABC$ be a triangle with centroid $G$. Let the circumcircle of triangle $AGB$ intersect the line $BC$ in $X$ different from $B$; and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$. Prove that $G$ is the centroid of triangle $AXY$.

Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.

The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$. (a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$. (b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Circles centered at $ A$ and $ B$ each have radius $ 2$, as shown. Point $ O$ is the midpoint of $ \overline{AB}$, and $ OA \equal{} 2\sqrt {2}$. Segments $ OC$ and $ OD$ are tangent to the circles centered at $ A$ and $ B$, respectively, and $ EF$ is a common tangent. What is the area of the shaded region $ ECODF$? [asy]unitsize(6mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0); pair A=(-2*sqrt(2),0); pair B=(2*sqrt(2),0); pair G=shift(0,2)*A; pair F=shift(0,2)*B; pair C=shift(A)*scale(2)*dir(45); pair D=shift(B)*scale(2)*dir(135); pair X=A+2*dir(-60); pair Y=B+2*dir(240); path P=C--O--D--Arc(B,2,135,90)--G--Arc(A,2,90,45)--cycle; fill(P,gray); draw(Circle(A,2)); draw(Circle(B,2)); dot(A); label("$A$",A,W); dot(B); label("$B$",B,E); dot(C); label("$C$",C,W); dot(D); label("$D$",D,E); dot(G); label("$E$",G,N); dot(F); label("$F$",F,N); dot(O); label("$O$",O,S); draw(G--F); draw(C--O--D); draw(A--B); draw(A--X); draw(B--Y); label("$2$",midpoint(A--X),SW); label("$2$",midpoint(B--Y),SE);[/asy]$ \textbf{(A)}\ \frac {8\sqrt {2}}{3}\qquad \textbf{(B)}\ 8\sqrt {2} \minus{} 4 \minus{} \pi \qquad \textbf{(C)}\ 4\sqrt {2}$ $ \textbf{(D)}\ 4\sqrt {2} \plus{} \frac {\pi}{8}\qquad \textbf{(E)}\ 8\sqrt {2} \minus{} 2 \minus{} \frac {\pi}{2}$

Find the integer which is closest to the value of the following expression: $$((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2$$

Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.

Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$ (I. Gorodnin)

Let $ a,b,c,d$ be rational numbers with $ a>0$. If for every integer $ n\ge 0$, the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \text{Cannot be found} \qquad\textbf{(E)}\ \text{None}$

Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).

Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both $a/b$ and $b/a$), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.

Find all positive integers $n$ such that $4^{n}+2007$ is a perfect square.

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

If $a>1$, $b>1$, and $p=\frac{\log_b(\log_ba)}{\log_ba}$, then $a^n$ equals $\textbf {(A) } 1 \qquad \textbf {(B) } b \qquad \textbf {(C) } \log_ab \qquad \textbf {(D) } \log_ba \qquad \textbf {(E) } a^{\log_ba}$

Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Consider in the plane the network of points having integral coordinates. For lines having rational slope show that: (i) the line passes through no points of the network or through infinitely many; (ii) there exists for each line a positive number $d$ having the property that no point of the network, except such as may be on the line, is closer to the line than the distance $d.$

For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix is the largest $k$ such that there is a $k \times k$ submatrix with nonzero determinant.) One may note that \begin{align*} M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{array}\right) \\ M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ -1 & -1 & 1 & 1 & 0 \end{array} \right). \end{align*}

Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.

Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will also end up in a [i]fight[/i]. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights?

There are $n$ coins of the radius $r$ on a table which is a circle of the radius $R$. It is known that: a) any two coins either touch each other or have no common points; b) there is no place for one more coin on the table. Prove that \[ \dfrac12 \left(\dfrac{R}{r} - 1\right) < \sqrt{n} < \dfrac{R}{r}.\]