A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

Adam has a circle of radius $1$ centered at the origin. - First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces. - Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis. - Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle. What is the product of the lengths of all $18$ segments Adam drew? [img]https://cdn.discordapp.com/attachments/813077401265242143/816190774257516594/circle2.png[/img] [i]Proposed by Adam Bertelli[/i]

Let $a$ be a positive real number. Prove that a) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a$. b) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a$. (As usual, $\sigma(n) = \sum_{d\mid n} d$ and $\varphi(n)$ is the number of integers $1\le m\le n$ which are coprime with $n$.) Proposed by [i]Deniz Can Karaçelebi[/i]

Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$.

In a garden, which is organized as a $2024\times 2024$ board, we plant three types of flowers: roses, daisies, and orchids. We want to plant flowers such that the following conditions are satisfied: (i) Each grid is planted with at most one type of flower. Some grids can be left blank and not planted. (ii) For each planted grid $A$, there exist exactly $3$ other planted grids in the same column or row such that those $3$ grids are planted with flowers of different types from $A$'s. (iii) Each flower is planted in at least $1$ grid. What is the maximal number of the grids that can be planted with flowers?

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed below. What is the least number of circles needed to make the total shaded area at least $2023\pi$?

Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.

9.6, 10.6 Construct for each vertex of the trapezium a symmetric point wrt to the diagonal, which doesn't contain this vertex. Prove that if four new points form a quadrilateral then it is a trapezium. ([i]L. Emel'yanov[/i])

Consider the $8\times 8\times 8$ Rubik’s cube below. Each face is painted with a different color, and it is possible to turn any layer, as you can with smaller Rubik’s cubes. Let $X$ denote the move that turns the shaded layer shown (indicated by arrows going from the top to the right of the cube) clockwise by $90$ degrees, about the axis labeled $X$. When move $X$ is performed, the only layer that moves is the shaded layer. Likewise, define move $Y$ to be a clockwise $90$-degree turn about the axis labeled Y, of just the shaded layer shown (indicated by the arrows going from the front to the top, where the front is the side pierced by the $X$ rotation axis). Let $M$ denote the move “perform $X$, then perform $Y$.” [img]https://cdn.artofproblemsolving.com/attachments/e/f/951ea75a3dbbf0ca23c45cd8da372595c2de48.png[/img] Imagine that the cube starts out in “solved” form (so each face has just one color), and we start doing move $M$ repeatedly. What is the least number of repeats of $M$ in order for the cube to be restored to its original colors?

How many seven-digit positive integers do not either start or end with $7$?

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \plus{} d,y \equal{} bx \plus{} c$ and $ y \equal{} bx \plus{} d$ has area $ 18$. The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \minus{} d,y \equal{} bx \plus{} c,$ and $ y \equal{} bx \minus{} d$ has area $ 72.$ Given that $ a,b,c,$ and $ d$ are positive integers, what is the smallest possible value of $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$

Let us consider $a,b$ two integers. Prove that there exists and it is unique a pair of integers $(x,y)$ such that: \[(x+2y-a)^{2}+(2x-y-b)^{2}\leq 1.\]

How many rectangles are in this figure? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0); B=(20,0); C=(20,20); D=(0,20); draw(A--B--C--D--cycle); E=(-10,-5); F=(13,-5); G=(13,5); H=(-10,5); draw(E--F--G--H--cycle); I=(10,-20); J=(18,-20); K=(18,13); L=(10,13); draw(I--J--K--L--cycle);[/asy] $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.

On the circle $\gamma$ the points $A$ and $B$ are selected. The circle $\omega$ touches the segment $AB$ at the point $K$ and intersects the circle $\gamma$ at the points $M$ and $N$. The points lie on the circle $\gamma$ in the following order: $A, \, \, M, \, \, N, \, \, B$. Prove that $\angle AMK = \angle KNB$. (Yuri Biletsky)